TREATISE
DIFFERENTIAL EQUATIONS.
SUPPLEMENTAL Y VOL UME.
N
PREFACE.
THE present volume contains all that Professor Boole •wrote for the purpose of enlarging his Treatise on Differential Equations. Had he lived to publish the second edition he would doubtless have incorporated his more recent investi- gations with the original work, and it is therefore necessary to explain why another plan has been adopted.
In some cases Professor Boole had indicated that certain portions of the original work were to be omitted and their places supplied from the manuscripts; but on examination it appeared that in subsequent passages of the work there were references and allusions to the portions thus marked to be omitted which would not apply to the substituted matter. Thus in attempting to carry out the directions it would have been necessary to accept the responsibility of making many alterations, and consequently to incur the risk of fail- ing in the attempt to improve the original form.
Moreover the Treatise had been for some time out of print, and the long delay which must have been caused by the labour of reconstruction would have produced serious inconvenience to students at Cambridge and elsewhere. Pro- fessor Boole himself was always especially anxious to consult
VI PREFACE.
the advantage of students, and those who had the charge of his manuscripts were naturally inclined to adopt a course of which they believed he would himself have approved.
The design of reconstructing the Treatise was therefore abandoned; and it was resolved that the original volume should be reprinted, and that the manuscripts should be collected and published separately. This plan has the ob- vious recommendation of enabling those who are already familiar with the original work to turn their attention readily to the new investigations. It will be seen that many of the Chapters of the present volume may be re- garded as independent essays or memoirs which lose nothing by being separated from the other volume ; and indeed no indications had been left by Professor Boole of the place which such Chapters were to occupy in the enlarged edition.
I have printed all the unpublished matter relating to Differential Equations which I found among Professor Boole's papers. In a few cases it will be seen that an investigation is incomplete ; such investigations have however been in- cluded in the volume, because I was unwilling that anything should be lost which so great a mathematician had written on a subject he had long and carefully studied.
I trust that no serious error will be found in the volume, and that any faults which may be detected will be excused on account of the nature and difficulty of the task that had to be performed. Many of the manuscripts had not been finally revised ; some of them were very obscure and had to be carefully and laboriously copied for the press. In general the equations were not numbered, and thus only
PEEFACE. Til
blanks occurred in place of references; this circumstance often caused great trouble and perplexity: I hope however that a satisfactory result has been finally attained.
I may state for the benefit of those who are conversant with the first edition of the original work that the theo- rem which in the present volume is cited as contained in Chap. II. Art. 1 will be found in Chap. IV. Art. 2 of the first edition: the change was made by the direction of Professor Boole's interleaved copy. It was judged conve- nient to number the Chapters in the present volume in con- tinuation of those in the original work.
All additions of my own are enclosed within square brackets. The sheets have been read by the Rev. J. Sephton, Fellow of St John's College, as well as by myself, and the volume is much indebted to his care and accuracy. Obvious mistakes in the manuscripts were of course corrected; thus, for example, the table at the end of the volume was calcu- lated by Mr Sephton, because the table in the manuscript was rendered erroneous by the use of a wrong sign in a formula.
I. TODHUXTEE.
ST JOHN'S COLLEGI, Nwtmber, 1865.
LIST OF PROFESSOR BOOLE'S WRITINGS.
In tlie Philosophical Transactions. On a General Method in Analysis, 1844, pages 225... 282.
On the Comparison of Transcendents, with certain applications to the Theory of Definite Integrals, 1857, pages 745. ..803.
On the Theory of Probabilities, 1862, pages 225.. .252.
On Simultaneous Differential Equations of the First Order in which the Number of the Variables exceeds by more than one the Number of the Equations, 1862, pages 437. ..454.
On the Differential Equations of Dynamics. A sequel to a Paper on Simultaneous Differential Equations, 1863, pages 485... 501.
On the Differential Equations which determine the form of the Roots of Algebraic Equations, 1864, pages 733. ..755.
In, llie Transactions of the Royal Irish Academy.
On the Analysis of Discontinuous Functions. Vol. 21, 1848, pages 124... 139.
On a certain Multiple Definite Integral Same Vol., pages 140. ..149.
In tJte Transactions oftlie Royal Society of Edinburgh.
On the Application of the Theory of Probabilities to the Ques- tion of the Combination of Testimonies or Judgments. Vol. 21, 1857, pages 597. ..653.
In tJie Bulletin de VAcademie...de St Peterslourg.
Consid6rations sur la recherche des integrates premieres des 6quations differentielles partielles du second ordre, Vol. iv. 1862, pages 198. ..215. [See page 143 of the present volume.]
In Crelle's Journal fiir Mathematik.
Ueber die partielle Differentialgleichung zweiter Ordnutjg Rr + Ss + Tt+U (s* - rt) = V. Vol. 61, pages 309. ..333.
LIST OF PROFESSOR BOOLE 3 WRITINGS. IX
In the Cambridge Mathematical Journal.
Researches on the Theory of Analytical Transformations, with a special application to the Reduction of the General Equation of the Second Order. VoL 2, 1841, pages 64... 73.
On Certain Theorems in the Calculus of Variations. Same Vol., pages 97... 102.
On the Integration of Linear Differential Equations with Con- stant Coefficients. Same VoL, pages 11 4... 11 9.
Analytical Geometry. Same Vol., pages 179. ..188.
Exposition of a General Theory of Linear Transformations. VoL 3, 1843, pages 1...20, 106. ..119.
On the Transformation of Definite Integrals. Same VoL, pages 216. ..224.
Remarks on a Theorem of M. Catalan. Same Vol., pages 277. ..283.
On the Transformation of Multiple Integrals. VoL 4, 1845, pages 20.. .28.
On the Inverse Calculus of Definite Integrals. Same VoL, pages 82... 87.
Notes on Linear Transformations. Same Vol., pages 167. ..171.
On the Theory of Developments. Same VoL, pages 2 14... 223.
In the Cambridge and Dublin Mathematical Journal.
On the Equation of Laplace's Functions. VoL 1, 1846, pages 10. ..22.
On the Attraction of a Solid of Revolution on an External Point. VoL 2, 1847, pages 1 ... 7.
On a certain Symbolical Equation. Same VoL, pages 7... 12.
On a General Transformation of any Quantitative Function. Vol. 3, 1848, pages 112. ..116.
The Calculus of Logic. Same VoL , pages 1 83 ... 1 98.
On a General Theorem of Definite Integration. Vol. 4, 1849, pages 14... 20.
On the Theory of Linear Transformations. Vol. 6, 1851, pages 87. ..106.
X LIST OP PKOFESSOE BOOLE'S WEITINGS.
On the Reduction of the General Equation of the ntYl Degree. Same Vol., pages 106... 11 3.
Letter to the Editor of the Journal. Same Vol., pages 284, 285.
Proposed Question in the Theory of Probabilities. Same Vol., page 286.
On Reciprocal Methods in the Differential Calculus. Vol. 7, 1852, pages 156. ..166, and Vol. 8, 1853, pages 1...24.
In the London, Edinburgh, and Dublin Philosophical Magazine... Third /Series.
Remarks on the Rev. B. Bronwin's Method for Differential Equations. Vol. 30, 1847, pages 6... 8.
ISTote on a Class of Differential Equations. Same Vol., pages 96, 97.
Remarks on a Paper by the Rev. Brice Bronwin, On the Solution of a particular Differential Equation. Vol. 32, 1848, pages 413. ..418.
Remarks on a Paper by the Rev. Brice Bronwin, On the Solu- tion of a Particular Differential Equation. Vol. 33, 1848, page 21 1. Notes on Quaternions. Same Vol., pages 27 8... 280.
In the Fourth Series of the same Magazine.
On the Theory of Probabilities, and in particular on Mitchell's Problem of the Distribution of the Fixed Stars, Vol. 1, 1851, pages 521... 530.
Further Observations on the Theory of Probabilities. VoL 2, 1851, pages 96. ..101.
An Account of the late John Walsh of Cork. In a letter from Professor Boole to Professor de Morgan. Same Vol., pases 348.. .358.
Solution of a Question in the Theory of Probabilities. Vol. 7, 1854, pages 29... 32.
Reply to some Observations published by Mr Wilbraham in the Philosophical Magazine, Vol. 7, p. 465, on the Theory of Chances developed in Professor Boole's ' Laws of Thought.' Vol. 8, 1854, pages 87.. .91.
LIST OF PEOFESSOR BOOLE'S TVEITISGS. XI
On the Conditions by which the Solutions of Questions in the Theory of Probabilities are limited. Same Vol., pages 91... 98.
Further Observations relating to the Theory of Probabilities in reply to Mr Wilbraham. Same Yol., pages 175, 176.
On a General Method in the Theory of Probabilities. Same YoL, pages 431... 444.
On certain Propositions in Algebra connected with the Theory of Probabilities. Yol. 9, 1855, pages 165... 179.
On a Question in the Theory of Probabilities. By A. Cayley, Esq. [This paper embodies some observations by Professor Boole.] Yol. 23, 1862, pages 361... 365.
On a Question in the Theory of Probabilities. Yol. 24, 1862, p. 80.
Separate Publications.
An Address on the Genius and Discoveries of Sir Isaac Newton. Lincoln, 1835.
The Right Use of Leisure. London, 1847.
The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning. Cambridge, 1847.
The Claims of Science. London, 1851,
An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. London, 1854.
The Social Aspect of Intellectual Culture. An Address de- livered in the Cork Athenseum — Cork, 1855.
A Treatise on Differential Equations. Cambridge, 1859.
A Treatise on the Calculus of Finite Differences. Cambridge, 1860.
[This list contains all Professor Boole's writings which have fallen under the notice of the editor ; it is possible that there may be a few omissions.]
CONTENTS.
CHAPTER
XIX. ADDITIONS TO CHAPTER II
XX. ADDITIONS TO CHAPTER VII. ..... 7
XXL ADDITIONS TO CHAPTER VIII ....... 9
XXII. ADDITIONS TO CHAPTER IX ....... 38
XXIII. ADDITIONS TO CHAPTER X. ..... .46
XXIV. ADDITIONS TO CHAPTER XIV. ..... eg
XXV. ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFER- ENTIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSO-
CIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 74
XXVI. HOMOGENEOUS SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL
EQUATIONS ......... 90
XXVII. OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS Off THE
FIRST ORDER ........ 96
XXVIII. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND OBDEB 119
• XXIX. ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION Sr+Ss+ Tt + U(s2-rt) = V, IN WHICH K, S, T, IT, V ARE
GIVEN FUNCTIONS OF x, y, z, p, q ..... 145
XXX. ADDITIONS TO CHAPTER XVII. . . . . . 175
XXXI. THE JACOBIAN THEORY OF THE LAST MULTIPLIER . . 200
XXXII. THE DIFFERENTIAL EQUATIONS OF DYNAMICS [FRAGMENT] 218
XXXIII. ON THE PROJECTION OF A SURFACE ON A PLANE . .221
I
CHAPTER XIX.
ADDITIONS TO CHAPTER II.
1. [!N Chapter n. Art. 9, two methods are given for solving the differential equation
(ax + by + c} dx + (ax + I'y + c) dy = 0.]
But there exists another transformation by which the equa- tion may be reduced to, (because it may be constructed from), an equation in which the variables are separated.
Assume as this equation
(Ay + C] dx'+ (A'x'+ C') dy = 0 ...... (1)
and let x — x + w^y, y = x + m^y.
It will be seen that in these equations united we have as many constants as in the original equation. Now on substi- tuting in the assumed equation the values of x and y', and comparing with the equation given, we deduce a system of relations equivalent to the following, viz.:
The quantities m^ , m^ are roots of the quadratic am* — (b + a) m + V = 0.
The quantities A, A', C, C' are determined by the system of equations
A + A' = a, C+C' = c,
^ 4 A'm^ = a, Cm^ + C'm^ = c' B.D.E. II.
IS
2 ADDITIONS TO CHAPTER II. [CH. XIX.
from which we find
— a cm, — c
A'=
m9-m1 m^-m,
am, — a cm, — c
— m.
Now (1) gives on dividing by (Ax' + C') (Ay + C} and integrating
-^ log (A'x' + C'} + \ log (Ay + (7) = const.,
* ' -^*-
or (A'x' + C'} T (Ay1 + C}2 = const.,
which on substitution and reduction gives
i {(am1 — a'} (x + m1y] + cml — c'^r"'
{(amz — a) (x + mzy) + cmz — c}ami-a'
2. Under certain circumstances the general solutions of differential equations of the first order fail. This happens in the above example if m2 = m1, the solution then reducing to
1 = const.
The theory of the deduction of the true limiting form of the solution in such cases requires a distinct statement.
Let the supposed general solution be represented by u=C,
C being the arbitrary constant and u a function of x, y, and constants which are not arbitrary. Suppose too that when one of these constants k assumes a particular value «, the function u reduces to a constant v. Then we have
u — v C — v
/C ~~ /C /C *~~ /C
Now the second member being a function of an arbitrary constant is equivalent to an arbitrary constant and may be
ART. 2.] ADDITIONS TO CHAPTER II. 3
replaced by C. The first member is a vanishing fraction, the limiting value of which is (-wj, the brackets being used to
denote that after the differentiation k is to be made equal to K. Hence the solution becomes
In applying this theory to the reduction of the general solution (2) in the case in which m1 = mt , it must be observed that the numerator of the first member is the same function of ml , x, y, as the denominator is of m^ , x, y ; or attending solely to their functional character with respect to wi1? w2, we may affirm that the numerator is the same function of 7nt as the denominator is of m£. Representing these func- tions by ^(mj, <j>(m2) respectively, we have
But m1, ra2 being roots of a quadratic equation may be represented in the form
ml = m + k, mt = m — k, the roots becoming equal when k = 0. Hence
<f> (m + k]
u =
— — ; .
<f> (m — k]
Therefore since
m + k} d<j> (m — k) _ dfy (m — k
dk dm dk dm
we have
- k} dk - -
1—2
ADDITIONS TO CHAPTER II. [CII. XIX.
7 j / - 5 - — - 5 --
. ,, fdu\ r dm dm
therefore -77
dkj
(7 Thus the solution "becomes on putting C for — ,
40
or --- Jog |7ayn _ a'\ (x + my\ _j_ cm _ c'l = C.
dm am — a
3. [The next Article seems to have been intended to ap- pear in the enlarged form of Chap, n.; but I cannot discover what precise position it would have occupied. I conjecture that " the above demonstration" refers to Chap. II. Arts. 2, 3; and I have accordingly supplied a reference to equation (3) of Chap. ii.
I had myself drawn Professor Boole's attention to Chap. II. Arts. 2, 3. The geometrical process of Chap. n. Art. 3, ap- pears to have been first given by D'Alembert in his Opus- cules, Vol. iv. p. 255. D'Alembert calls it a demonstration; it seems to me only an illustration, at least in the brief form of the text : and that such was Cauchy's opinion may perhaps be inferred from the elaborate investigation given by Moigno, to which Professor Boole refers in Art. 5 of the present Chapter.
I had also drawn Professor Boole's attention to the state- ment at the end of Chap. n. Art. 12, that only one arbitrary constant was involved. Accordingly Article 5 of the present Chapter developes this statement, and Article 4 sesms intended to bear on the same subject.]
4. In the above demonstration the relation between y and x is regarded as one of pure magnitude, and the interpreta- tion of the differential equation becomes a limiting case of that of the equation of finite differences (Eq. (3), Chap. n.). But if we represent x and y by the rectangular co-ordinates
ART. 5.] ADDITIONS TO CHAPTER II. 5
of a moving point on a plane the differential equation may be interpreted directly. For supposing it reduced to the form
we see that the direction of motion is constantly assigned as a function of the co-ordinates of position. The entire motion is therefore determinate as soon as the initial point is fixed. The result of the motion is a line or curve wholly continuous or subject to irregularities according to the nature of the func- tion f(x, y). That the arbitrariness of origin is geometri- cally equivalent to the appearance of a single arbitrary con- stant in the relation connecting x and y may be shewn thus.
Let y = $(•*<>, y*,*}
be the relation between x and y indicated by the supposed motion, x0, ya being the initial point of departure. Then this point being on the line of motion, x0. y0 are particular values of a; and y, so that we have from the above equation
which establishes a relation between x0 and y0, and shews that there exists virtually but one arbitrary constant.
5. It is proved in Art. 3, Chap, n., that the constants a*0, y0, initial values of the variables x, y in the solution of the differential equation of the first order, are necessarily equivalent to one arbitrary constant. I shall shew from the form of the above solution that this a priori condition is actually satisfied.
Developing the expression for y [see Eq. (30) of Chap, n.] in ascending powers of cc, we have
........ (32)
the summation extending from n = r to w = cc. Formin
6 ADDITIONS TO CHAPTER II. [CH. XIX.
lience the differential coefficients of Ar with respect to cr0 and y0, and reducing by (28), we shall find
. dAr , , . dAr
whence in particular
y\ dA« = A
Eliminate "between these equations f1(x0, ?/0), and we have
Therefore, by Prop. I., Ar is a function of A0, so that tlie solution reduced to the form (32) contains but the single arbitrary constant A0.
It remains to notice that the solution must be applied only under the conditions of convergency, i.e. under the con- dition that the ratio of the wtb to the (n — l)th term tends to a limit less than unity as n tends to infinity. For a discus- sion of the failing cases of this test see ' Finite Differences,' Chap. v. Generally it is desirable, in order to secure rapid convergency, to divide the interval x — <r0 into separate equal portions, to each of which the general theorem of solution may be applied. If x — x0 be very small the theorem may be approximately represented by
y -#o=/0»e >300-*o) •
On these principles Cauchy has founded remarkable methods of solution, which deserve attention from the commentary on the limits of error on their application by which they are accompanied (Moigno, Vol. II. pp. 385 — 434).
CHAPTER XX.
ADDITIONS TO CHAPTER VII.
1. [THIS Article relates to Art. 2 of Chap, vn.]
The sense in which (9) may "be said to constitute the general solution of the differential equation is this. We obtain from it
giving any particular value to C this will geometrically represent a curve consisting of two branches, and giving to C every possible value we obtain an infinite system of such curves, each consisting of two branches. The aggregate of branches thus obtained is evidently the same as the aggre- gate of curves given by the two primitives (5) and (6), un- restricted by any connexion between cl and c2. In this sense then the solution (9) is general, that it includes all the parti- cular relations between y and x which are deducible from the original primitives (5) and (6). And it is only in this sense not general that it groups these relations together in a particular manner.
To the expression of the complete primitive a certain variety of form may be given without affecting its generality in the sense above affirmed. Thus, if to the solutions of the component differential equations we give the forms
ye^-c^O, logy + ax-cz=0,
we should have, by the same procedure, as the expression of the complete primitive,
(ye™ — c) (log y + ax - c] = 0,
8 ADDITIONS TO CHAPTER VII. [CH. XX.
an equation which may equally with (9) be regarded as the complete primitive of the differential equation given, and which in geometry represents the same totality of branches of curves as (9), with this difference only, that they are differ- ently paired together.
2. [This Article relates to Art. 3 of Chap, vn.]
The question will here naturally arise, Since if F= c be a solution of one of the component differential equations, f(V) = c, in which f (V] denotes any function of F, is also a solution, by Chap. IV. Art. 3, why not give to the complete primitive the form
or the stricter form
in which /^FJ, ,^(F2), ...y»(Fn) denote arbitrary functions of Fj, F2,..., Fn respectively — stricter because the presence of arbitrary constants and functions in the previous form is a superfluous generality? It is replied that though the form just given is analytically more general than (15), it is not more general than (15) with such freedom as is permitted in the interpretation of the arbitrary constants. In a physi- cal or geometrical application we should not only be per- mitted to assign a particular value to the arbitrary constant in (15), so deducing what in reference to its source would then be termed a particular primitive, but to combine the re- sults of different determinations of c together, so as to obtain every form of solution which is implied either in the func- tional equation (F}> or in its component primitives
V —c V = c V —c
r 1 *"! > 'a — c 2 ' ' ' » ' » ~ °n •
The same considerations justify us in speaking of (15) as the complete primitive, and not as a complete primitive.
CHAPTER XXI.
ADDITIONS TO CHAPTER VIII.
1 . [THE Singular Solutions of Differential Equations of the First Order received great attention from Professor Boole, and the Chapter devoted to that subject is one of the most valuable and important in his work. He continued his re- searches after the publication of his first edition, and intended to reconstruct the Chapter with great improvements in the second edition. After carefully examining the manuscripts I came to the conclusion that it would be very difficult to re- write this portion of the work so as to connect the old matter with the new ; and thus it seemed best to reprint the original Chapter vin. with corrections of obvious misprints, and to print the matter intended for the revised form in the present volume. The plan gives rise to some repetition; but this seems unimportant, compared with the advantage of preserv- ing in the author's own language all that he left on an in- teresting and important point which he had carefully Studied.
2. It may be of service to the student to reproduce the substance of some remarks on his Chapter vin. which were sent to Professor Boole soon after the publication of his first edition ; tor there is evidence in his manuscripts that he paid great attention to such remarks while engaged in the revision of his work, and thus the reason and the meaning of some of his additions and changes may be made more obvious. These remarks will occupy the next Article,
3. The two pages beginning with " And these conditions are sufficient/' and ending with "do not lead to conflicting
10 ADDITIONS TO CHAPTER VIII. [CH. XXT.
results" forming part of Arts. 3 and 4 of Chapter vin., seem obscure and difficult. The following may perhaps be substi- tuted with advantage.
The only ways in which
dy ^ df(x, c] and dy = df(x, c} df(xt c) dc dx dx dx dx dc dx
can be equivalent when c is variable, are
fi\ -u df(x-> c) A (1) when -' ;=Q,
/^N i . c)
(2) when -'^ ;=co;
in the latter case -~ = GO , and therefore -=- = 0. and this ax dy
implies that the singular solution is of the form x = constant. Thus there can be no singular solutions except such as
df (x c]
are found from — ---,-- — - = 0, and such as are found from dc
x = constant.
Similarly, if the complete primitive be expressed in the form x = P(y, c), there can be no singular solutions except
such as are found from - -j2 — = 05 an<i sucn as are found
dc
from y = constant.
In Art. 8 of Chapter vin. we read, " We may pass over the case in which the above equation is satisfied independ- ently of c, because the relation obtained would involve x
{JT)
only, while it is a condition accompanying the use of -f- = oo
that it leads to solutions involving y at least." It is ob- jected, Why may we pass over this case? Such a case might occur and furnish a solution, and then we should want to know the character of that solution. Take for example
p = xny ; here if n is negative, -j- is infinite when x = 0, and
y ^n + I
this is a singular solution. For the general solution isy=cen+l,
ART. 3.] ADDITIONS TO CHAPTER VIII. 11
and so x=Q is not a case of it. The words — ichile it is a condition... at least — seem very difficult, for by supposition
we are now investigating what is furnished by -~ = oo .
Professor Boole met the objection in substance thus :
" It will be found that the rules in the book are correct in this case. "What is implied in the Chapter, though not stated
with sufficient clearness, is that if -f- = oo leads to a solution
dy
which does not involve y in its expression, nothing is to be inferred whether it is singular or not. Then the proper test is
" 1'
\pj
i - =co. ax \
In this example we have
-Z. = co gives x* = GO ; no inference ; dy
dx\~>= y
Hence x =- 0, provided n is between 0 and — 1 , or y = 0.
Consider these separately :
First. Let n be between 0 and— 1, and x = 0. This is by the test a singular solution. Substituting it in the com- plete primitive we get y = c, which confirms this.
Second. Let y = 0. This satisfies the differential equa- tion; but from the fact that it comes from -7- (-] = oo we
ax \p)
have no inference ; from the fact that it does not come from -±- = GO we have the inference that it is a particular integral : it corresponds to c = 0.
12 ADDITIONS TO CHAPTER VIII. [oil. XXI.
There remains the case of x=0 when n is between — 1
and — co . As this does not satisfy -=- ( - ) = co , we infer that * J dx \p)
it is a particular integral. To prove this we have
When x = 0 this gives, since 1 +. n is negative,
c = co or c = — co ,
according as y is positive or negative. This is like Ex. 2 of Chap. vin. Art. 8."
The remark made by Professor Boole in the above reply, that if — = GO leads to a solution which does not involve j
nothing is to be inferred... is important. It corrects the state- ment put too strongly in Chap. vill. Art. 7, " All we can affirm
is that if ~- = co gives a solution at all it will be a singular solution."
From Art. 8 onwards it seems assumed that a solution for which — = 0 is always to count as a singular solution, even if
it should coincide with a particular integral. This does not seem to have been quite the view of the former part of Chap- ter vni. : see Arts. 5 and 6 of the Chapter.
In Ex. 3 of Art. 9 we read, " the second is obviously a singular solution." This means that since we have a solu-
tion which makes -f- infinite, we conclude that it is a singular ay
solution.
So in Ex. 5 of Art. 11 we read, " is evidently a singular solution," when it seems better to say, " and is therefore a singular solution."
4. The additional matter relating to Chapter vm. begins with another example which was to be placed at the close of Art. 3 of that Chapter.]
ART. 4.] ADDITIONS TO CHAPTER VIII. 13
Ex. The differential equation
has for its complete primitive
\x* + y* — m* — y — c = 0.
T-T d(h y dd> x
Here y- = . y — - 1, -f- = (
/HI »/^ I „.* »v,» /7-T? i/^." I ^.
<7y V JJ +y*—m* dx *Jx* + y* — m*
Hence -- = -- - = - - -
/ , ,
- V J72 -f -
771
Both -^ and — vanish then if dc dc
st»+ /-*?=*<>.
This therefore is the singular solution and it satisfies "both the tests, as both x and y are contained in its expression.
Of the partial tests
d4> d<b dd>
__ — O !— — m — -r>
^ v> J — J J — *' l
dc dx ay
the first is not satisfied, the last two are satisfied.
The determination of c as a function of x by the solution
df(x c\ of the equation J ^ — = 0 is equivalent to determining
what particular primitive has contact \vith the envelope at that point of the latter which corresponds to a given value of x.
One important remark yet remains. The elimination of c between a primitive y=f(x, c) and the derived equation dy
— = 0, does not necessarily lead to a singular solution in the < c
14 ADDITIONS TO CHAPTER VIII. [CH. XXL
sense above explained. For it is possible that the derived equation
dc
may neither on the one hand enable us to determine c as a function of x, so leading to a singular solution ; nor, on the other hand, as an absolute constant, so leading to a particular primitive. Thus the particular primitive
dtj being given, the condition -~- gives
eex = 0,
whence c is + co if a; be negative, and -co if a? be positive. It is a dependent constant. The resulting solution # = 0 does not then represent an envelope of the curves of particu- lar primitives, nor strictly one of those curves. It represents a curve formed of branches from two of them. It is most fitly characterized as a particular primitive marked by a sin- gularity in the mode of its derivation from the complete pri- mitive.
All the foregoing observations and conclusions may be extended to the case of solutions derived from the condition dx _ dc
5. We have seen that the equation -~ — 0 may be satisfied
by an absolutely constant value of c, so leading to a particu- lar primitive and not a singular solution. In this case -v/r (x + h, c) as well as i|r (x, c) would vanish, and the nume- rator of (9), instead of being the difference of a finite and an infinite quantity, would be the difference of two infinite and equal quantities. [Sec Chap. vill. Art. 8.] It would not there-
fore be infinite. Hence we conclude that -*- would not become
ay
infinite for a particular primitive in the strict sense of that
ART. 5.] ADDITIONS TO CHAPTER VIII. 15
term, i. e. for a solution derived from the complete primitive by giving to c an absolutely constant value.
This is one point of contrast between the conditions *y_0 ^=ao
-, - VT, 7 - vAJ •
dc ay
There is another not less important. As the numerator of (9) may become infinite not only when -fy (x, c) = 0, but also when ty (x, c} = infinite, we see that a relation between
y and x which makes ~ infinite will not necessarily satisfy
the differential equation. On the other hand, it is not a par- ticular primitive in the strict sense of that term.
dx Exactly in the same way the condition -^ = 0, as relating
to the complete primitive, leads to the condition
d (\_ -
as relating to the differential equation, with the same points of difference in the respective applications.
dti m~l
Ex. Let -2- = myn , and suppose m a positive constant
greater than 1.
dp -
Here ^=(m -!)/»,
which becomes infinite when y = 0. As this involves y and satisfies the differential equation it is a singular solution.
To confirm this conclusion we may refer to the complete primitive
y=(x-cr,
which does not give y = 0 for any particular value of c.
Now let m be a positive constant less than 1. We have still — = co when y = 0 ; but this value of y no longer satis-
16 ADDITIONS TO CHAPTER VIII. [CH. XXI.
fies the differential equation. It is not a solution at all, nor would it result from the application of the condition ^ ;- = 0
to the complete primitive. The distinction of character of the two tests is here made manifest.
6. We may express the most important results of the foregoing investigations in the following theorem.
THEOREM. Every solution of a differential equation of the first order which is derived from the complete primitive by giving to c a variable value will, if it involve y in its expres- sion, satisfy the condition
*-«;
dy
and if it involve x, the relation d /I
-r- 1-1=00.
ax
But relations satisfying these conditions will not neces- sarily be solutions of the differential equation.
In applying this theorem the following points must be carefully attended to.
1st. No conclusion can be drawn from the satisfying of
the condition ~ = GO when the relation in question does not
dy contain y in its expression, nor from the satisfying of
d /IN
7T ~ =co ax \pj
when the relation in question does not involve x in its ex- pression. For these conditions being respectively derived
from -if- = 0 and - ~ = 0 are subject to the same limitations dc dc
in their application.
2ndlv. It may be that -f- or -=- ( - ) assumes for a particu- dy dx \pj
lar relation between x and y the indefinite form n . In this
*
ABT. 6.] ADDITIONS TO CHAFiEll VIII. 17
case we must seek by the development of its terms or by other known modes its true limiting value or values. Finite values will indicate particular primitives, infinite values sin- gular solutions, and when such values emerge together out of the same relation between the variables, the solution will be a particular primitive possessing the geometrical properties of a singular solution. Its locus will be a particular curve en- veloping other curves of the same family.
See Examples 2 and 3 of Chap. vin. Art. 11. We have seen that the conditions
iln d f\\
77- = QO» j- [- =:0 ay dx \j)J
indicate in general the existence of a relation between c and x or c and y. And when that relation is such as to enable us to determine c as a continuous function of one of the vari- ables, the corresponding solution of the differential equation is singular, and is geometrically represented by an envelope of the curves of primitives. But it may be, as we have seen in a particular example, that the relation does not determine c as a function of x or y ; but according to the language already used, c is a dependent constant, or in some other way different from the constant of an ordinary particular primitive. Let us examine in particular instances the kind of singularity which may hence arise.
Ex.1. Given
Here -f- = - (1 + log y).
dy x ^ ° dl
This becomes infinite if x = 0 ; but this not involving y must be rejected. Again, it becomes infinite if y = 0, and this proves to be a solution of the differential equation, the limiting value of the indeterminate function in the second member being 0 (Todhunter's Differential Calculus, Chap. x.). Xow the complete primitive is y — e", discussed in Art. 4. The constant c is there shewn to be dependent, the solu-
B. D. E. II. 2
18 ADDITIONS TO CHAPTER VIII. [cil. XXI.
tion y = 0 emerging from the complete primitive by making c = — co if x be positive, and c = co if a; be negative.
Ex. 2. Given f-^J — xy ~- + y2logy=Q.
xii + ?/ (x'2— 4 loe: y)b Here p = -^^ — — - j
3fl
^P _ •'K ± (»2 — 4 log g)_ j. 1
therefore =
. , ,,
(a-8 -4 logy)4
and this is made infinite by y = 0 and by a;2 — 4 log T/ = 0, i.e. by
Both satisfy the differential equation. Now the complete primitive is
y = <?*-*.
We see at once therefore that the second of the above solu- tions is singular. The first however is deducible from the complete primitive by making c = co or c = — co , irrespec- tively of the sign or value of x, provided only that x be finite ; not so however if x be infinite. The value of c is not therefore in the most absolute sense independent of that of x. If from the complete primitive we seek the singular solution
by the condition :J = 0, we get the two equations
The second of these determines c as a function of x, and leads to the second of the solutions obtained above. The first, though it does not determine c as a function of a*, still ex- presses a relation between c and x, which is the ground of the fulfilment of the condition
dp
/ =00. dy
ART. 6.] ADDITIONS TO CHAPTER VIII. 19
We may further notice a peculiarity arising from this rela- tion. Supposing x finite and the solution y = 0 a particular integral, it presents the singularity that it is the only case in which two particular integrals agree. We might in any com- plete primitive, by changing c into c2, get two values of c for the same particular integral, but then it would be for every particular integral.
One negative character seems indeed to mark all the cases in which a solution involving y in its expression satisfies the
condition -^ = <x . It is that such solutions do not emerge dy
from the complete primitive by the attributing of a single and absolutely constant value to c. The relation which makes •— infinite satisfies the differential equation only because it satis- fies the condition -f- = 0, and this implies a connexion be- ac
tween c and x, which is the ground of a real though it may be unimportant singularity in the solution itself.
At this point, then, the question arises, whether the term singular solution shall be confined to that class of solutions, the loci of which represent the envelopes of curves of primi- tives, or shall be extended to all solutions which, satisfying the
condition -~ = co , indicate the existence of a relation be-
dy
tween c and x, and possess an actual singularity arising from this source. While the all but universal consent of mathe- maticians is in favour of the former course, it is to be remem- bered that the question is solely one of definition. Xot such is the question how singular solutions of the envelope species, or as would more generally be said true singular solutions, are to be distinguished from all other solutions. This we now propose to consider. The question is not an isolated one. It stands in close relation to a series of properties of singular solutions which admit of an orderly development.
2—2
20 ADDITIONS TO CHAPTER VIII. [CII. XXI.
Discrimination of singular solutions of the envelope species.
7. A negative test, which in the great majority of cases suffices for the present object, is suggested by the following consideration.
dit The differential equation determining -£- as a function of
d*y dsi/
x and y determines also ~~ , -rjj , ad inf., and the know- ledge of these enables us to construct in a developed form the complete primitive. See Chap. n. Art. 12.
fit I Ci tl
The values of -jr- , -r4 » &c. ad inf., as derived from the
CtJfr C13C
differential equation, are the same as those derived from the complete primitive.
But a solution deduced from the condition -j- = <x> is only
ay
7 9
constructed so as to yield the same value of -y- as the given
differential' equation does. If it be of the envelope species, the curve it represents has in general no continuous contact with the curve of any particular primitive. It will not there- fore generally yield the same values for , ••{ , -7-^ , &c. as
the differential equation does. It will not therefore generally satisfy the differential equations of an order higher than the first, which would be derived from the given equation by dif- ferentiation. Hence we have the following Proposition.
PROPOSITION. If a relation which makes -J- infinite satisfy
ay
the given differential equation of the first order, but do not satisfy all the higher differential equations obtained from it, such solution will be singular and of the envelope species.
ART. 7.] ADDITIONS TO CHAPTER VIII. 21
Ex. 1. By comparison with its complete primitive we saw in Art. 5 that -^- = my *" has for a singular solution y = 0 when m is a constant greater than 1.
We will first suppose m a fractional quantity greater than 1, and endeavour to deduce the character of the solution with- out making use of the complete primitive.
From the solution we have
g-O, g-0,*c. «?,»/.
But from the differential equation
g-(— i)^|-«{«-i)^,
and generally
<Fy m~T
~jr = m(m-l)...(m-r + l)y'*.
Hence, when r is less than m, the substitution of y = 0 gives
as before. But if r is greater than m, it gives
d'y
^ = CO'
We conclude that the solution is of the envelope species. Secondly, suppose m a positive integer greater than 1.
In this case we find, when r is less than m, the same series of values as before • but for r = m we have
and this also shews the solution to be of the envelope species.
22 ADDITIONS TO CHAPTER VIII. [CH. XXI.
Ex. 2. The differential equation
!+/ is satisfied by
Is this a singular solution or a particular integral ? From the solution we find
dy x d2y _ 4 dx ~ y' dx2 y* '
From the differential equation we shall have
dx2 2 (y - xp)
ClI/
substituting in which the value of -jr- , obtained from the proposed solution, we find
da? 2y* y3 '
Now this differing from the value before obtained, we con- clude that the solution is singular and of the envelope species.
And this result is verified by comparing the solution with the complete primitive
As the test above exempl fied is merely negative, it is in- sufficient. For it is conceivable that an enveloping curve should have an infinite order of contact with each of the curves which it envelopes, and this is also possible. Any test found- ed upon a comparison of the values of differential coefficients, any test therefore furnished by the Differential Calculus, would be insufficient for the discrimination of such cases.
Ex. 3. Given -j- = y (log y)\
ART. 8.] ADDITIONS TO CHAPTER VIII. 23
Here -^- = 00 gives y = 0, and this satisfies the differen- tial equation.
From this solution we find
From the differential equation we have
which consists of y multiplied by a rational and entire func- tion of logy. It is easy to see that all the higher differential coefficients of y hence derived will possess the same character. And all such vanish with y.
We can therefore neither affirm nor deny that the proposed solution is of the envelope species.
8. Before demonstrating a general Rule for the discrimi- nation of solutions of this character, we shall notice certain of their properties which serve to indicate in what direction the Rule is to be sought. [See Chap. vili. Art. 14.]
As the exact differential equation differs from the sup- posed given differential equation by having acquired a factor which the singular solution makes infinite, so the given dif- ferential equation may be said to differ from the correspond- ing exact one by containing a factor which the singular solu- tion makes to vanish. If we knew that factor, we could by rejecting it reduce the given differential equation to a form in which it would no longer be satisfied by the singular solution. Now Poisson has shewn on a particular assumption, which does not however affect the principle of the demonstration, that this factor can be found when the singular solution is known. His demonstration is in substance as follows.
Let us represent the given singular solution of the dif- ferential equation by
w = 0, u being a given function of x and y. Then introducing u and
2i ADDITIONS TO CHAPTER Till. [fH. XXI.
x instead of y and x as variables, the differential equation after transformation will assume the form
du ... .
5 -/(<*>«).
Now this equation being satisfied by w = 0 and the first member vanishing, the second must also. Poisson now assumes, and the assumption must be carefully noted, this second member to be capable of being developed in ascending positive powers of u. Supposing it so developed, the diffe- rential equation becomes
in which A, B,... are functions of x, and a, /3,... ascending positive indices.
Hence if u = 0 be a singular solution we have, putting p „ du
f°r^'
-/- = Jaw-1 + Spue-* + &c. = « . an,
But this demands that there should be at least one nega- tive power of u in the development in the second member. Therefore a — 1, the lowest index, must be negative. There- fore a being already positive must lie between 0 and 1.
"We may give therefore to the transformed differential equation the form
du _
a being a positive fraction, and Q not vanishing with w. Hence, dividing by M°,
du
ART. 9.] ADDITIONS TO CHAPTER VIII. 25
a differential equation which is not satisfied by u = 0, since u = 0 gives w1"" = 0, and the first member vanishes while the second member does not vanish. In its present form then the equation is not satisfied by u = 0. We see also that the property of being satisfied by u = Q has been lost not in reality through a transformation, but through the rejection of an algebraic factor ua from the transformed equation. It has been shewn in the treatment of Clairaut's equation, how in the ascent by differentiation to an equation of a higher order a somewhat analogous effect is produced, the singular solu- tion emerging out of a factor of that higher equation.
If we inquire what is essential in Poisson's demonstration, we shall find it to consist in that the transformed equation is of the form
in which while Q neither vanishes nor becomes infinite when u = 0, the functions
both vanish with u. The question whether U is of the form ua as Poisson supposes, or is not, is wholly immaterial . This will fully appear from the demonstration of the follow- ing theorem, which is in effect Poisson's freed from arbitrary assumptions.
9. PROPOSITION. If u = Q be a solution of a differential equation of the first order between y and x, and
du . . fa =/(*,«)
represent the form which that equation assumes when u and x are assumed as variables instead of y and x, then if f(x, u) be resolved into two factors Q, U, of which Q neither vanishes nor becomes infinite when u = 0, while the functions U and
-jj both vanish when u = 0, then the differential equation can be reduced to a form in which it shall cease to be satisfied by
26 ADDITIONS TO CHAPTER VIII. [CH. XXI.
In the statement of this proposition x is supposed to be constant in the integration relative to u.
The differential equation after the transformation which introduces u and x as variables becomes
£-«*
Let
so that v is in general a function of x and u, the form of which is known by integration when that of U is given. And again, transform the differential equation by making v and x the variables instead of u and x. We have
fdv\ _ dv dv du \dxj dx du, dx'
in which -=- is the differential coefficient of v with respect to
(1 'V
x, on the above hypothesis as to the constitution of v as a
function of x and u. while IT ) is the differential coefficient
\dxl
on the hypothesis that v is reduced to a function of x alone by the conversion of u into a function of x.
dv 1 du
omce ~T = ^TTI ~T~
(i?/ i J ci'jr
the above equation becomes
Now if u = 0 give v = 0 for all values of x, it will there- fore give
-*
and further,
^w_ _J p ^f_A dx~dx)0 (J~Q>
ART. 9.] ADDITIONS TO CHAPTER Till. 27
since we are permitted to make u = 0 before effecting the differentiation with respect to x. Hence the equation re- duces to
0=0.
And this is not satisfied, since by hypothesis Q does not vanish with u.
Hence if u = 0 make I -~ = 0, the transformed differen- tial equation will no longer have u = 0 for a solution.
COR. Assuming Q = 1, which does not violate the hypo- thesis respecting Q, and gives
U=f(x, u),
we see that if
— = f(x it]
be satisfied by u = 0, and if at the same time u = 0 gives
du _0
(x M) ~~
the differential equation can be transformed so as to cease to admit of the solution u = 0.
It is obvious however that it is best to assume Q so as to make the subsequent integration for determining v the sim- plest possible.
It is manifest that a solution which can thus be made to cease to satisfy the differential equation cannot be a particular primitive. For the complete primitive of the transformed differential equation which it does not satisfy is convertible into the complete primitive of the original differential equa- tion which it does satisfy, merely by writing therein for v its expression as a function of x and y. It cannot therefore be a case of the complete primitive in any sense. It must be a singular solution of the envelope species.
The converse proposition still remains to be proved.
28 ADDITIONS TO CHAPTER VIII. [CH. XXI.
10. PROPOSITION. If u = 0 be a singular solution of thf> envelope species of a differential equation of the first order, and if by assuming u and x as the variables, the differential equa- tion is reduced to the form
du j.. .
T*=^x' «)•
then will
du
,/(»,*)
become 0 when u = 0.
Let the complete primitive be represented by
F(x, u) = C, then, since
dF(x, u) dF(x, u) du _ dx du dx
we have if for brevity we represent F (x, u) by F,
du dx
therefore *
f(x,u)
dx
Now w = 0 being a singular solution, F(x, 0) is not a con- stant ; for if it were, the complete primitive would, on giving to C the constant value in question, yield u = 0 as a particu- lar primitive. And this would equally be the case whether that constant were finite or infinite in value. We see then
that F(x, 0) must be a function of x, and therefore --A— ' — -
must either be a function of x, or a finite constant differing from 0 ; the latter if F(x, 0) be of the form ax + b, the former
ART. 10.] AUDITIONS TO CHAPTER VIII. 29
dF(x, u) if it be not of that form. Therefore the value of -
dx
when M = 0, since in this we are permitted to make u = 0 before differentiating with respect to x, will be a function of JL-, or a finite constant differing from 0.
Now it is manifest that in general
' 1 dF j TrTdF ,
— — — a u = H\ -j- du, at du J0 du
dx
0
where H is some value intermediate between the greatest and
least values which —TV assumes within the limits of intesra- ar
dx
tion. When these limits are, as in the above case, infinitesi- mal, we have
jr ^ o .
Hence
dx du
But we have seen that — —^ — - does not vanish. Hence
its reciprocal, the first factor of the right-hand member of the above equation, does not become infinite. Again,
F(x, u)-F(x, 0) vanishing when u = 0, we have
du
when u is made infinitesimal as was to be shewn.
30 ADDITIONS TO CHAPTER VIII. [CH. XXI.
It will "be observed that the previous general expression for I -j- . becomes infinite if u = 0 is a particular integral.
* 0 J \ ' **/
For then, F(x, 0) being a constant, - ' vanishes, while
GLJC
F(x, u) —F(x, 0) does not vanish so long as u differs by however small a quantity from 0.
These propositions form the ground of the following Rule for the discrimination of singular solutions of the envelope species from all others.
11. RULE. The proposed solution being represented by u = 0, let the differential equation, transformed by making u and x the variables, be du
Determine as a function of x and u the integral
du
in which U is either equal to f(x, u), or to f(x, u) deprived of any factor which neither vanishes nor becomes infinite when u = 0. If that integral tend to 0 with u the solution is singular.
Ex. 1. Determine whether y = 0 is a singular solution or particular integral of the differential equation
Here, since u — y, no preliminary transformation is needed.
r du i
We have ., -y ... = — —
•/0y(loS*/) iogy
which tends to 0 with y. Hence the solution is singular. To verify this we observe that the complete primitive is
ART. 11.] ADDITIONS TO CHAPTER VIII. 31
and this cannot be reduced to y = Q by giving any constant value to c.
We have seen in Art. 7 that the test -which is founded upon the comparison of differential coefficients does not suffice to characterize the above solution.
Ex. 2. The equation ~ - = — — is satisfied by y = 0.
CLJC> J.
Is this solution singular or particular?
Here also no transformation is required. We have, reject-
ing the factor - which neither vanishes nor becomes infinite x
when =
[
— — = log log y — log log 0 , los: v
and this being infinite, however small y may be. may properly be said to tend to infinity as y tends to 0. The solution is therefore particular.
It will perhaps appear at first sight as if in the above ex- ample we ought to write
when y is made equal to 0. But the course of the demonstra- tion shews that the value of the definite integral must be first obtained on the hypothesis that u (in this case replaced by y] is finite, and then the limiting value which its expression approaches to, as u approaches to 0, be sought. And in this •case, since for all finite values of u however small the integral is infinite, its limiting value is infinite.
The complete primitive in the above case is
y = f,
and the nature of the solution y = 0 has already been dis- cussed in Art. 4.
32 ADDITIONS TO CHAPTER VIII. [cil. XXI.
History of the Theory of Singular Solutions,
12. It is remarkable that while the theory of enveloping curves and surfaces was at once founded and developed by Leibnitz in 1692 — 4*, the corresponding theory of the singular solutions of differential equations has been of very slow growth. The existence of these solutions was first recognised in 1715 by Brook Taylor; it was scarcely more than recognised by Clairaut in 1734. Euler, in a special memoir, entitled Expo- sition de quelques Paradoxes dans le Calcul Integral, published in the Memoirs of the Academy of Berlin for 1756, n'rst made them a direct object of investigation ; but the foundations of their true theory were only laid in 1768 in his Institutiones Calculi Integralis. Laplace, Lagrange, Legendre, Poisson, Cauchy, and De Morgan have in various ways developed and extended that theory; but there has been so remarkable a want of unity and connexion in this long series of researches, that important portions of the theory appearing in a too isolated form have been neglected, forgotten, and rediscovered. I purpose here to give a brief account of what seems most cha- racteristic, rather than of what is most original in their several researches ; for the germs of nearly all subsequent discoveries on the subject are to be found in the great work of Euler.
Taylor and Clairaut appear to have been led by accident to the noticing of singular solutions ; the former while directly occupied on the solution of differential equations, the latter while discussing a remarkable class of problems relating to the connecting properties of different branches of the same curve. Taylor gave them the name singular, while Clairaut, and Euler too in his memoir, regarded them as a species of paradox, not merely from their non-inclusion in the general integral, but from the mode of their discovery through a process of differentiation. The memoir of Euler, though it sheds no light on the real nature of these solutions, contains
• Ada Eruditorum, 1692, p. 168 ; 1694, p. 311. Opera, Tom. in. pp. 2G4, 296.
Methodus Jncrementorum, p. 26.
Mtmoirea de CAcadcmie des Sciences, 1734, p. 209.
ART. 12.] ADDITIONS TO CHAPTER VIII. 33
an interesting theorem concerning their connexion with the form of the differential equation, viz. If this equation can be brought to the form
Vdz=Z(Pdx+Qdy],
in which z is a function of x and y, and Z of z, then will
Z=0
be a singular solution. In his Institutiones Calculi XktegraKs, Tom. i. p. 393, however, Euler gives a rule which is the counterpart of that of Cauchy. [See Chap. vin. Art. 12.] He shews that if u = 0 be a particular integral, and if the differential equation be reduced to the form du , .
, du
then
I
* o
d> (x, u]
The limits of integration are here supplied. The reasoning, which is not fully developed, is the following. From the transformed equation we have
du
„ i du Hence x =
<j> (x, u) '
x _ If du
~7r ~ * + "TV
If this be satisfied by a solution involving x and y, and if that solution be a particular integral, then on putting for x its value in terms of u and integrating, the above equation will be satisfied by giving some particular constant value to C. But if the supposed particular integral be u = 0, then x and M being independent, we may perform the integration with respect to u as if x were constant. The resulting equa- tion cannot be free from x unless C be infinite, and then it
B. D. E. n. 3
34 ADDITIONS TO CHAPTER VIII. [CH. XXI.
evidently not "be satisfied unless I — c be infinite.
J <l>(x, u)
We infer then that this is a necessary condition in order that u = 0 may be a particular integral.
This is Euler's fundamental theorem, and from this, by means of an hypothesis agreeing with that of Poisson con- cerning the form of the transformed differential equation, he arrives at the condition
dp
— = &. dy
[In the passage to which Professor Boole refers, Euler does not undertake to discuss the nature of any solution, but only of a solution of the form x = constant. On his page 408 Euler proceeds to discuss the nature of any solu- tion. Professor Boole seems to me to attribute too much to Euler. For the convenience of those who wish to ex- amine the original, I will give the reference to the passages in the later editions of Euler's Institution's Calculi Integralis : Vol. I. pages 343 and 355 of the edition of 1792 ; Vol. I. pages 342 and 354 of the edition of 1824.]
Laplace in the Memoirs of the French Academy for 1772, p. 343, established the tests
dp d fl\
-f- = co , 3- - = co ,
dy dx \pj
and shewed their respective uses. He established also the test which consists in the comparison of differential coefficients, and he supposes it universal. He adopts the hypothesis of his predecessors as to the forms of expansion, but with some recognition of its insufficiency.
Lagrange in the Memoirs of the Academy of Berlin for 1774, p. 197, and 1779, p. 121, appears first to have developed the theory of singular solutions in its two forms of derivation from the complete primitive and derivation from the differen- tial equation, and to have established the essential connexion of these. But supposing the differential equation to be ex- pressible in the rational form
ART. 12.] ADDITIONS TO CHAPTER VIII. 35
and employing the differential coefficients of F(x,y,p) in- stead of those of p he was led to sacrifice rigour to symme- try. One of his results has often since been adopted as a test of singular solutions. It may be thus stated.
PROP. A singular solution makes the general value of
cPu
-=5 , deduced from the differential equation in its rational and
air
integral expression, to assume the form - .
[The demonstration is given in Chap. vill. Art. 14.]
This ambiguity of value of -^ is evidently but an expres- sion of the fact that the contact of a curve of the complete primitive and that of the singular solution is not in general of the second order.
The result given in equation (5) of Chap. vill. Art. 14 has also been adopted as the test of singular solutions.
The researches of Poisson and Cauchy have already been noticed. It is certainly remarkable that the final test to •which Cauchy's analysis led should be essentially the same as that which had been discovered by Euler so long before.
Professor De Morgan has thrown an important light upon the nature of the conditions
dp dp
j =00» ;/ =GC> dy ax
which are fulfilled by all singular solutions in the expression of which x and y are both involved. He has shewn that any relation between x and y which satisfies these conditions will
73
satisfy the differential equation unless it make -~ , as derived
from the differential equation, infinite ; that it may satisfy the
ji
differential equation even if it make -7 \ infinite ; lastly, that
CUE
3—2
36 ADDITIONS TO CHAPTER VIII. [CH. XXI.
if it do not satisfy the differential equation, the curve it represents is a locus of points of infinite curvature, usually cusps, in the curves of complete primitives.
The proof is as follows : Let p = $ (x, y]
be the differential equation. Then the proposed conditions are
d$ (x, y] d<f> (x, y)
—fry- ' ~^~ '
therefore by differentiation,
__ ---
dxdy dy2 dx dx* dxdy dx
whence we have
dy _ dx dy _ dxj dx
dy* dx dy
These are two equivalent expressions for the same value of -,— . The question now is, under what circumstances this
CtiC
value of ~ will satisfy the differential equation.
Cm?
Now from that equation we have by differentiation
d*y _ d<$> d<f> dy dx* dx dy dx'
whence
d*y d<f> ily _ dx* dx dx d$
dy
ART. 12.] ADDITIONS TO CHAPTER VIII. 37
If th infinite,
If then -T-? be finite we have, since fr and -j- are both dx1 dx dy
dy _ dx
dy
and this by the rule for the evaluation of fractions of the
form ^ is equivalent to the value in either of its forms before
obtained for -/-. Hence, any relation which satisfies the ax
given conditions and makes -~ finite, will satisfy the diffe- rential equation.
And the same result holds even if —^ be infinite, provided
that ~~ -r- -3T- vanish. aar dy
Lastly, as when this result does not hold, the failure is due
72
to the infinite value of -^ , we see that the line in which the
locus of the proposed relation intersects the curves of primi- tives will be a locus of their points of infinite curvature.
[Transactions of the Cambridge Philosophical Society, Vol. ix. Part H.]
Legendre's ^Memoir of 1790 throws but little light upon the subject of this Chapter. But it exhibits the theory of the singular solutions of differential equations of the higher orders, both ordinary and partial, in a form of great beauty, and will be noticed in the proper places.
( 38 ) .[CH. xxii.
CHAPTER XXII.
ADDITIONS TO CHAPTER IX.
1. BY successive application of the second theorem of Chap. IX. Art. 13, a linear equation of the ntb order may be reduced to one of the (n — r)th order, if r distinct integrals of what the given equation deprived of its second term would be are known.
The reduction may however be effected immediately by the method of the variation of parameters. In this and in most general investigations connected with differential equa- tions great advantages in point of brevity and of the power of expression are gained by the employment of the symbol of summation S, and of the language of determinants. I shall exemplify this here.
Suppose the given equation to be
and let yl} ya,'»yr be r particular values of y, satisfying the equation
Thus y = c^ + c^ . . . + cryr
is a solution of the latter equation including these particular solutions. We shall represent this by
(3), and regarding the quantities clt c2,...cr, represented here by
ART. 1.] ADDITIONS TO CHAPTER IX. 39
cf as variable parameters, shall seek to determine them so that the above value of y may satisfy the equation given.
These r parameters, enabling us to satisfy t — 1 arbitrary conditions, besides satisfying the differential equation, we may choose these so that
dy cPy cTly dx' da?*'
may be the same inform as if ct, ca, . . . cr were constant. Xow from (3)
whence
provided that the condition
be satisfied. Differentiating the first of these equations, we find in the same way that
provided that the condition
2 d?h dci _ 0 dx dx
be satisfied. And thus continuing we see that the system of r equations
will hold true provided that the r — 1 conditions
40 ADDITIONS TO CHAPTER IX. [CH. XXII.
be satisfied. In each of these equations the symbol £ is to be interpreted by giving to i the successive values 1, 2,... r, and taking the sum of the results.
Differentiating the last of the equations (4), we have
ty. = s & 4- * ^y* — *
dxr idxr ' * dx'-1 dx'
As we cannot impose the condition that the last term of this equation shall vanish, let z represent its unknown value, then
%-*%+> ..................... ««•
Now the system of equations (5), together with
constitute a system of r simple algebraic equations deter- mining by solution the r quantities
dc^ dc2 dcr dx' dx'"' dx
in terms of their coefficients and of z, and therefore in terms of x and z, since the coefficients are known as functions of x. It is evident also that as the second members of all the equations but one vanish, and the second member of that is z, the values so determined will be of the form
X^X^... Xr being known functions of x. Thus the r un- known quantities /y1v -y^ are made to depend upon only
one unknown quantity, viz. z. It remains then to deter- mine z.
For this purpose we must complete the expression of the differential coefficients of y, and substitute in the given dif- ferential equation, and then seek to satisfy that equation.
ART. 1.] ADDITIONS TO CHAPTER IX. 41
Xow differentiating (6) we have
i V <fyi <ki , dz
1 • djJ dx "*~ dx
<c on substituting for -7-' the value XjZ as above determined.
(••V
We observe that the coefficient of z is here a known function of a\ If we differentiate this equation and in the result sub-
dc
stitute as above for -, , we shall have a result of the form dx
L and J/ being known functions of x. Ultimately then we have
* * n K~T
Q
dx* l dx* dx ' ' dx" '
Thus, while y and the differential coefficients of y up to the (r — I)01 are of the same form as if c,, cs,... cr were constant, the succeeding ones differ in containing an additional portion consisting of z, and differential coefficients of z multiplied by known functions of x. The result of substitution of these values in the given differential equation will therefore consist also of two classes of terms, viz: terms under the sign of summation, which will be the same in form as if cx, c2,...cr were constant, and terms involving the differential coefficients of z up to the (n — r)th, with multipliers which are known functions of x. We shall in fact have
^-r^ +
42 ADDITIONS TO CHAPTER IX. [CH. XXII.
Now yi being by hypothesis an integral of (2), the first line of the above equation vanishes, and there remains the linear equation of the (n — r)th order
Supposing z hence determined, we have in general
r
and hence
y = yAXlzdx + yAX.izdx +yr\XTzdx,
and as z will have n — r distinct values, each involving an ar- bitrary constant, the above equation will furnish n — r distinct values of y, each involving an arbitrary constant. It is to be observed that no arbitrary constant need be added in the inte- gration of the terms X& dx, for the effect of such addition would only be to reproduce the known integrals c^. In this way, however, the equation would represent the general integral of the differential equation given.
2. Let us examine the form of the result in the particular case in which r=n — l.
Here we have
dxm' ldxn from m = 0tom = n— 2, then
ART. 2.] ADDITIONS TO CHAPTER IX. 43
Accordingly the differential equation for z will be
Now the equations for determining
dx
<
become on putting Xp for 4p , and writing for brevity y\ for
Whence, by the theory of determinants,
^ Mfyf*''
J/ standing for the determinant
«
2/3
(*-«)
44 ADDITIONS TO CHAPTER IX. [CII. XXII
is ultimately a dr*y{\ I dM
Now the determinant is ultimately a function of x ; and such indeed that
'daTlJ M dx ' For
i, ^y
'
_ ___
dx ** dyt dx dyl dx ' dy£n^ dx .
Now M being homogeneous and of the first degree with respect to the quantities y1, y2, ..... yjt_2, we have
Hence 2 -=— y- is what M becomes when in its expression
V\->y*i •••yn_i are changed into y/, 7/2', ... y'n_v therefore it is what M becomes when two of its rows of elements become identical ; therefore it vanishes. In like manner all the other sums in (8) vanish excepting the last, for y^n~l\.... y,^n~l] is not a row of elements of the determinant M. Thus we have
^ dM (n_1}_
Hence
, 1 dM
Thus the equation (7) becomes dz /I dM
therefore z = -i €~fA^ f Me^** Xdx.
M J
ART. 2.] ADDITIONS TO CHAPTER IX. 45
Hence, since
dci_ -,?• _ 1 = iZ =
whence
v f1 dM j we have y = % J ^ ^^ 2^r,
2 being given above.
In the case of X = 0, vre have
whence
( 46 ) [CH. xxiii.
CHAPTER XXIII.
ADDITIONS TO CHAPTER X.
1. THE theory of singular solutions of differential equations of the higher orders has been presented in the most complete form which it has yet received by Legendre. (Memoires de V Academie Eoyale des Sciences, 1790, p. 218.) He determines first the possible forms of these solutions considered as emerg- ing from the complete primitive by the variations of its arbi- trary constants, and secondly the theory of their derivation from the differential equation itself. I shall follow the same order, and shall in the end endeavour to point out in what respect Legendre's theory may be regarded as complete, and in what respect it is imperfect.
Suppose the differential equation to be of the nih order, and let it when solved with respect to the highest differential coefficient of y be represented by
yn=$(x>y>yny*T~y^ .................. (i),
in which, for brevity,
dy d?y dny
ni — ". ni — _ !? ni — - tL
2/1 dx> *• dot'"'** XT'
Let also its complete primitive, solved with respect to y, be represented by
y=f(x, alt oa,...ow) ..................... (2),
ap o2, ... an being the arbitrary constants of the solution. If we differentiate (2) with respect to x, regarding ax, a2, ... an no longer as constants but as functions of x, so to be deter- mined as to leave the expressions for y^ y2, ... yn as functions
ART. 1.]
ADDITIONS TO CHAPTER X.
47
of al, a.,. ... «„ the same as before, we shall have, on repre- senting the second member of (2) by/,
•whence
df df da^ df da, df dan
dx dav dx da, dx " dan dx '
provided that
df dal df <Jan df dan _
Differentiating on the same hypothesis the first of these two equations, we find in the same way
I U' J
y* = ^?'
provided that
d?f dal d^f da, d?f dan
dx da. dx dx da, dx" dx da dx
1 z n
And continuing thus, it results that the system
._£ ,=#..,»#.. ..f«
dx ' dx^ ' dx*
will be satisfied, i.e., that yl, y,, . . . yn will have the same ex- pressions when at, a2, ... an are variable as they have when these are constant, provided that the law of their variation be determined by the conditions
da. dx da. dx " da. dx
JL Z
* J- *^ 2 i • * n ___ A
da^ dx dx da, dx dx' ' dan dx dx
da_i d"f da, d*f dan _ dx da. dx*~l dx ' ' da. dx"'1 dx J
48 ADDITIONS TO CHAPTER X. [CH. XXIII.
In this system the coefficients of
oa1 daz dan
dx ' dx ' dx
are known functions of x, a^ , a2 , . . . an when the form of f is known.
Eliminating
dal da2 dan dx ' dx ' ' dx '
we have a relation between x, ax, a2, . . . an; and this relation, with the given complete primitive and the first n — 1 of the derived and reduced equations, viz., with
_/ -df
will enable us to eliminate a1? «2, ...«„, and to obtain a rela- tion of the form
dy d2y dn
This is a differential equation of the (n — l)th order. It dif- fers in its origin from the given differential equation, in that a new relation between x, al5 a2, . . . an has been employed in place of the nih equation, derived by differentiation from the complete primitive, for the elimination of the constants.
The differential equation of the (n — l)th order thus obtained has an integral expressing y in terms of a1, and n — 1 arbi- trary constants. This is the most general form of a singular solution of the differential equation.
It is possible that the elimination of al , a.2 , . . . an may lead to a resulting differential equation which, instead of being of the order n — 1, is of the order n — 2, n — 3, &c. The complete integral of such equation would be a singular solution of the differential equation. These possible types of solutions are distinguished by Legendre according to the number of arbitrary constants which they contain. A solu-
ART. 1.] ADDITIONS TO CHAPTER X. 49
tion containing n — 1 arbitrary constants is called by him a singular solution of the first order ; one containing n — 2 ar- bitrary constants a singular solution of the second order, and so on.
Adopting this language we might term the complete primi- tive a singular solution of the order 0.
Lastly, any relation between x and y, which satisfies the given differential equation, will constitute a particular case, either of the complete primitive or of one of the general forms of singular solutions above defined. In the case of differential equations of the first order it is seen that no arbi- trary constant can appear in the expression of the singular solution.
Ex. The equation
V^-a.
3) (dx X
has for its complete primitive
ax* y=— + bx+a* + b* .................... (6),
required its singular solution.
Proceeding as above, we find on the hypothesis of a and b being variable parameters, the same formal expressions for
-~ , -r^ as if those parameters were constant, viz.
dx
•(7),
provided that the variation of a and b be such as to satisfy the conditions
da db
dx dx
B. D. E. II.
50 ADDITIONS TO CHAPTER X. [CH. XXIII.
Eliminating hence -j- and — , we have
a; 2a - — - 2bx = 0.
2
And from this, the complete primitive, and the first of the derived equations (7) eliminating a and b, we find
This is the differential equation of the first order, by the solution of which the most general form of the singular solu- tions of the given differential equation will be determined.
Reducing it to the form
x .
and integrating, we find
This then is the general expression for the singular solu- tions of the given differential equation. We see that it in- volves in its expression one arbitrary constant.
The differential equation (9) may properly be termed a sin- gular first integral of the given differential equation. The singular first integral (9) has itself also a singular solution, viz.
1 . 1 4
rt/ — __ _ sy*£ __ _ /W* •
y- 4* 1&x
but this is not a solution of the original differential equation. Nor have we any right to expect that it should be so. A singular solution of a differential equation of the first order does not necessarily satisfy the differential equations of higher orders derived from that equation, Chapter xxil. Art. 7.
2. It remains to establish the theory of the derivation of the singular solution from the differential equation without the mediation of the complete primitive.
ART. 2.] ADDITIONS TO CHAPTER X. 51
Resuming the differential equation in its reduced form (1), and representing its second member by <f>, suppose an infini- tesimal variation given to the arbitrary constants of its com- plete primitive, and let the symbol 8 be used to denote the corresponding derived variations of y, yl} . . .ya. Then we have
and so on. Hence, substituting and transposing,
d'Sy d* d-l8y d* <Z"-% . _ . .
~dx* ~dy^ "dx^ ~d^ ~d^ '
Let us consider the real nature of this equation.
If a value of y, suppose y = ^r(x), satisfy the given differ- ential equation, that value substituted in the coefficients
of the above equation •will convert them into functions of x, and the equation itself will become a linear differential equa- tion, the solution of which will determine $y as a function of x. If the differential equation (10) be really, as it is appa- rently, of the ?ith degree, §y will have n arbitrary constants, O, . . . a, and will be of the form
j, P8, . . . Pn being functions of x. Hence
p I
We see thus "that the given solution y=TJr(x) will be a articular case of this general integral involving n constants. t will therefore be a particular integral of the proposed.
4—2
52 ADDITIONS TO CHAPTEE X. [CH. XXIII.
If, owing to the constitution of its coefficients, the differential equation (10) be of the degree n — 1, we shall have
- n-i )
and y = ^r(x] will then be a particular case of a solution involving n — 1 arbitrary constants. It will therefore be a singular solution of the first order. Even so, if the differen- tial equation (10) be of the degree n — 2, y = ty(x) will be a singular solution of the second order. And generally, if the differential equation be of the r& degree, y = ^(x) will be a singular solution of the order n — r.
Kesuming the equation (10) it is evident that it cannot
be of the degree n — 1, unless -r-^- be infinite. For, dividing
dy^
by 7 , we have
__ d<f> ' dxn dxn-* C'
in which the first term does not vanish unless ~— be infi- nite. This then is the necessary condition for a singular solution of the first order. For one of the second order we must have in like mariner
d<j> d(j)
and so on.
It follows hence that to find the singular solutions of a differential equation of the nih order, we ought to differentiate
the equation, regarding y, -^ , -^ , &c. as varying through
the variation of the arbitrary constants, to form in this way a linear differential equation for Sy, to examine the conditions tinder which this equation reduces to the (n — l)th, or to a lower degree, and to examine whether the most general relation be-
ART. 3.] ADDITIONS TO CHAPTER X. 53
tween x and y which satisfies such condition, satisfies also the given differential equation. If so it may be regarded as a singular solution.
Resuming the last Example, viz. dy 1 ^d*y_ (
and operating with 8 we have
which reduces to a linear differential equation of the first order for determining By, provided that we have
dy
=0.
Eliminating -y^ from the given equation by means of this there results
dy
and we find on differentiating this that it does constitute a solution of the given equation. It is therefore a singular first integral of that equation. We see that it agrees with the result obtained under the same name in the previous Article, and the rest of the solution need not be repeated.
3. Upon Legendre's theory, and upon its results, the fol- lowing observations may be made.
1st. We learn from it that there may exist different
eneral forms of the solution of a differential equation of the
n01 order, viz. the complete primitive involving n arbitrary
54 ADDITIONS TO CHAPTER X. [CH. XXIII.
constants, and general forms of singular solutions containing fewer than n arbitrary constants. A solution y = ty (x) of unknown origin being given, we construct a differential equa- tion for determining By, and, solving it, form the expression for y + By, and from the number of infinitesimal arbitrary constants it contains, determine the nature of that general value of y of which the given value is a particular case. Now we are not to infer from this that the form of y + By will be the same as the general value of y in question. But we may infer that it will be a form to which that general value is reducible. And the actual reduction will be effected by expressing the general solution (as is always possible) in a form permitting its expansion in ascending powers of the arbitrary constants, and in the expansion making these con- stants infinitesimal, and rejecting all powers of them above the first. In fact, if
y=f(x, a,, a
2,
be any general form of solution which, when we assign to al} «2,...ar particular values (e.g. make them vanish) re- duces to
then we shall have
y + By = +(x) + Ba, + &%... + Bar,
the brackets denoting that after differentiation we make at , a2 , . . . ar vanish.
This is that limiting form of the solution which Legendre's method enables us to construct by the solution of a linear dif- ferential equation ; and the ground of the sufficiency of his method consists in this, that the infinitesimal quantities
Ba1} 8aa, ... Bar)
which are in fact the arbitrary constants of that solution, are equal in number to the arbitrary constants of the general unlimited solution, the nature of which is thus made known.
ART. 3.] ADDITIONS TO CHAPTER X. 55
2ndly. Legendre's tests for differential equations of the higher orders are in kind and effect analogous to the tests
dp d 1
-T-=00, -=- - = CO
ay ax p
for differential equations of the first order. They enable us to decide whether a solution possesses singularity, not whether it possesses the envelope species of singularity. The comple- tion of Legendre's theory would consist in the discovery of those further tests dependent upon integration which corre- spond to the test of Euler and Cauchy for differential equa- tions of the first order.
[CH. XXIV.
CHAPTER XXIV.
ADDITIONS TO CHAPTER XIV.
[Art. 1 was intended to follow Chap. xiv. Art 2.]
1. As the condition of dependence of functions of two variables is of fundamental importance in connexion with the theory of ordinary differential equations, so the generalized condition of dependence of functions of any number of vari- ables forms a fundamental part of the theory of partial differ- ential equations. This is contained in the following proposi- tion.
PROP. I. If
... un are functions of #,, a?2, ...#n,
but are as such so related that some one of them is expressi- ble as a function of the others, or more generally that there exists among them some identical equation of the form
F(u1,u,,...un)=0, ................... (1),
BO that as functions of a^, #2, ...#n they are not mutually independent, then, adopting the notation of determinants, the condition
Ci T flT (IT
dxv ' dxz ' " " cfo,,
C?Mn </Mn <?Mn
/-j.-v* ' /7-T* /T'T*
UUJ- UU/0 €*•**/-
= 0
•(2),
is identically satisfied. Conversely, if the above condition be identically satisfied, the functions ut , uz , . . . un are not mutu- ally independent in the sense above explained.
ART. 1.] ADDITIONS TO CHAPTER XIV. 57
First let it be noticed that the Proposition is but a general- ization of that of Chap. n. Supposing U and u to be two functions of x and y, the condition of their dependence is affirmed to be
dU dU
dx ' dy
du du
dx' dy
i. e. it is the result of eliminating dx, dy, from the equations
dU
rlv. -L.
dx
= 0,
du
du dy
and therefore it is
dU du dU du _ dx dy dy dx
as expressed in Chap. n.
We proceed to the general demonstration.
Let the first member of (1), considered as a function of MI? M2, ... un be represented for brevity by F; then differen- tiating, we have
dF , dF .. dF ,
-j- du. + -7— du + ... + -j— dun = 0, du^ dua dun
from which it follows that if du^ dua, ... dun_1 are equal to 0, then is dun equal to 0; or, since MI} w2, ... un are functions of a?,, «2, ... acn, that if
du
du
du
du
dx,
+
\-^dx = 0 dx* "
^<-i j , 1 dx. = 0
(3),
58 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
then is
Thus the last n equations, linear with respect to dxlt dx2, ...dxn)
are not independent, and therefore by the theory of linear equations the determinant of the system vanishes identically. Now this is expressed by the condition (2).
It remains to prove the converse, viz. that if the condition (2) be identically satisfied, the functions ult u2, ...un will not be mutually independent.
First, the n — 1 functions w , ua, ... u^ are either mutually independent or not mutually independent.
If not, then the n functions ul , w2 , . . . un are not mutually independent, and the Proposition to be proved is granted.
If ult w2, ... un_i are mutually independent as functions of a?!, fl?2, ... xn, they may be made to take the place of n — 1 of these quantities, e.g. xt, o?a, ... xn_1 in the expressing of un, i.e. we may, by means of the expressions for w1? w2, ... w^, eliminate from that of un the quantities xl , x2 , . . . xn_1 , and so express un as a function of wz, w2, ... un_v and xn. Suppose this done, then the system (3), (4) will be converted into
dut = 0, du3 = 0, ...... dun-i — ^>
dun , dun , du , du ,
Now, the determinant (2) vanishing, the equations of the linear system (3), (4) are not independent ; therefore those of the transformed system, as written above, are not independ- ent; therefore the last equation of that system must be a consequence of the others which manifestly are independent. But from the form of that last equation we see that such can- not be the case unless we have
ART. 2.]
ADDITIONS TO CHAPTER XIV.
59
which implies that nn is a function of MI? uy, ...*Vi merely. Hence the functions ut. uz, ... ua are not independent, as was to be shewn.
The first member of the equation of condition (2) is com- monly called the functional determinant of MI} ua, ... un with respect to a^, x3, ... ccn. The proposition may therefore be expressed as follows.
The condition of dependence or independence of any sys- tem of functions of as many variables is the vanishing or non-vanishing of the functional determinant of the system.
On account of the great importance of this proposition it is desirable to illustrate it by an example.
Ex. Are the functions
x + 2y + z, x — 2y-r3z, 2xy — xz+ tyz - 2z* mutually independent or not ?
The equation of condition is
1, 2, 1
1, -2, 3 =0,
that is,
- 4 (— x + 4y — 4z) + 8 (2y - z) — 2 (2x + 4z) = 0,
which is identically satisfied. Hence the functions are de- pendent. In fact, representing them by u} v, w, we have
[Art. 2 was intended to follow Chap. xiv. Art. 4.] 2. As it has been shewn that a primitive
(1)
leads to a linear partial differential equation of the form
(2),
60 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
provided that u = a, v = b, are integrals of the system of ordi- nary differential equations
dx _ dy _ dz . .
7~~Q~ ti-
lt is evident that we shall obtain a solution of the partial dif- ferential equation (2) by constructing the system of ordinary differential equations (3), deducing their general integrals
u = a, v = 5, and then constructing from these the primitive (1).
But the question arises, Will this be the most general solu- tion of the partial differential equation given ?
That it will be so, may be shewn by means of the general proposition. See Art. 1.
For let w = 0 represent any solution whatever of the given partial differential equation. Differentiating this with respect to x and y, we have
dw dw dw dw
substituting the values of p and q formed from this in the given equation, we have
which must be identically satisfied.
In like manner, u = a, v = b being solutions of the same equation, we find
du du -du
dx dy dz which must be identically satisfied.
AP.T. 3.] ADDITIONS TO CHAPTER XIY. 61
Eliminating P, Q, E from these three equations, it results that the functional determinant of 'w, w, v, with respect to x, y, z, •will identically vanish. Hence w is a function of M and v, and the equation w = 0 is a particular case of
F(u, v) = 0,
•which is thus shewn to be the general integral of the given equation.
"We are thus led to the following general Eule.
RULE. To integrate the equation Pp+Qq=R we must form the system of ordinary differential equations
dx dy _dz ~P = ~Q~R'
deduce their general integrals in the form
u = a, v = b, and construct the equation
• F(u, v} = 0.
This will be the general solution sought. [Art. 3 was intended to follow Chap. xiv. Art. 5.]
3. The above theory may be extended to linear partial dif- ferential equations of the first order, without regard to the number of the variables.
First, the theory of the genesis of such equations is ex- pressed in the following proposition.
PROP. A primitive equation of the form
^(Ml,u,,...«=0 (1),
in which wt , vs , . . . un are any given functions of the vari- ables z, dependent, and a:,, a;,, . . . xn independent, will satisfy the linear partial differential equation obtained by eliminating dz, dx:J dxa) ... dxn from
62 ADDITIONS TO CHAPTER XIV. [CH. XXTV.
expressed as total differential equations with respect to the primitive variables, and the equation
dz -pl dxl — p2 dxa . . . — pn dxn = 0.
Of this important proposition I propose to give two distinct
proofs.
1st proof. Forming the total differential of the given equation we have, on representing its first member by F,
dF 7 dF 7 dF 7
-j— du. + -j- du, ... + 3— dun = 0. dul duz dun
Now this cannot be true for all forms of the function unless we have the separate conditions
Strictly to prove this, suppose F^ F^ ...Fn to be any n distinct and independent functions of ul,u2,...un, and as such, distinct and independent forms of F. Then the above equation gives
df\du df\du df\d dF0 , . dFn , dF0 ,
,d ^
Now Flt F?,...Fn being independent, their functional determinant with respect to W1,w2,...wn, does not vanish. This again is the condition necessary and sufficient that the above system of linear equations may be independent ; and this lastly being the case, their only possible solution will be
as was to be shewn.
ART. 3.] ADDITIONS TO CHAPTER XIV. 63
These equations in their developed expression
du. , du. , du. ^ du. 7
•T-I<£EI + — -dxy...+ j-Ldxn + -j-1 dz = Q,
CLU ClU -m f.
— cijc •••••••*••••••••*••*••••••• "T" ~i> — dz == \) y
dx. 1 dz
+ -^dz = 0,
enable us to determine the ratios of dxt, dx3, . . . dxn, dz in the form
dx1 _ dxt _ dxn _ dz ,9«
x^-x," -^-x"
where X15 -T2,...jrB,^ are functions of the original vari- ables. And now, forming the equation
and eliminating the differentials, we find
X^ + Xzp2 ...+ Xnpn = R for the partial differential equation sought.
2nd proof. Differentiating the given primitive with respect to xv as contained explicitly in the functions ut , u2, . . .un, and also implicitly in the same through z, we have, on represent- ing the first member of the equation by F,
dF (du. du\ dF (du. du\
- I - 1 J. /n _ * I _1_ - I - ? -|_ in - - I
du, \dxl +Pldz)+ du, (dx, +Pi dz)
dF /du du
+Pl= '
or
since
dF du, dF du, dF dun dF
L _L ^ I . ™ I _ ay — I)
du dx du dx ' ' du dx dz ^
dF du. dF du. dF dun dF
_ _ * l _. _ . i _ _ '* ^— _
dul dz du% dz " dun dz dz
64 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
Differentiating thus with respect to the remaining inde- pendent variables, we obtain finally the system
dF du^ dF_ du dx du
dFdu^ dF
* * * 1^ 7 7 i^ 7 r'l ~~ "j
aun ax, dz-^1
dF duv dF duq dF dun dF du, dx0 dua dx0 ' dun dx0 dz J
dF^dui dF du^ dF dun dF dul dxn duz dxn ' ' dun dxn dz **
from which, in combination with the equation dF du. dF dus dF dun dF
,_, ._. * l_ « I ™ — A
du^ dz duz dz ' dun dz dz we can eliminate
dF dF dF dF
du^ ' duz ' " " dun ' dz '
The result will be
du^ ch^ dun . =0,
ft If* /r/Y9 finf* ~^"
«-*»</*» VV«A/_, \A/VUy.
dUi d\ dun ^
dz ' dz'" ' dz '
or, converting rows into columns,
€tU+ CjuU^ dU+ CLlt*
I 1 I 1
dxl> dxz'" " dxn' dz
flu* dun dun dun
dxt' dxa'' " dxn ' dz
Pn Pv Pnt -1
ART. 3.] ADDITIONS TO CHAPTER XIY. 65
which is the determinant form of the result affirmed in the proposition.
The second of the above forms of demonstration seems to be preferable to the first,, in that it rests only upon the consi- deration of the one general form of the function F. I have, however, given the two proofs, chiefly in order to illustrate an important remark, viz. that, in nearly all general re- searches connected with partial differential equations of the first order, two modes of procedure, the one involving the use of differentials, the other that of differential coefficients, may be employed, and that between the forms to -which these respective modes give rise, a certain law of reciprocity will be found to exist.
The theory of the solution of the partial differential equa- tion
follows immediately from that of its genesis. If we repre- sent by
«! = «!> ttt=ail--M.= °.»
the integrals of the system of ordinary differential equations (2) a solution of the given partial differential equation will be represented by (1). That this will be also the most gene- ral solution may be shewn by the argument of Art. 1. For if w = 0 represent any solution, then since
dw dw dw dw
3^t
we find
Ydw
-a. j Sj- ...n-j -T-
1dxl Saor2 dxn dz
from which, in combination with the corresponding equations,
du d^ du^ ,Jfdul_
-A.J — r -A--J— ••• + -i* -j -- r -ft -j -- U, 1 dx * a«c dx dz
Y * j- Y * i.Y»A. 7?» - n
.A. j h -A -j— . . . + A, -= h 21 -j- = 0,
^dx aaa? dx dz
B. D. E. II.
66 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
eliminating Xlt Xz, ... Xn, R we obtain a result which ex- presses that the functional determinant of w, w1? ... un with respect to the original variables is virtually 0. Whence w is a function of w1? %2, ... un, and the proposed solution is in- cluded in the one to which the above method of solution leads.
That method may therefore be stated in the following Rule. RULE. To integrate the linear partial differential equation
X—+X — + X dz =R
1 dx^ 2 dx2 ' ' n dxn
form the system of ordinary differential equations dxl _ dxz _ dxn _ dz
and deduce their general integrals
then
F(Ul,u2,...un)=0
will be the general integral sought.
[The general observations were intended to follow Chap. xiv. Art. 6.]
General observations.
4. The relation which exists between a proposed linear partial differential equation and its auxiliary system of ordi- nary differential equations should be carefully studied. While it is proper to say as above that the general integral of the one requires the knowledge of all the integrals of the other, it is also proper to describe that general integral simply as the most general form under which an integral of the auxi- liary system can appear. If
ui = an W8 = os» ... wn = an are integrals of that system, then
ART. 5.] ADDITIONS TO CHAPTER XIY. 67
is the one general form of an integral of that system, and due regard being Lad to the arbitrariness of F, this is equi- valent to
5. The form which the auxiliary system assumes when the given partial differential equation is deficient in any of its terms should be noticed.
If Xl = 0, the auxiliary equation
becomes, on clearing of fractious,
dxl = 0. And thus, if Xt, Xt,... Xr vanish, the given equation being
X — + X ^Z + Y — = Y the auxiliary system will be
cLxr+. ctxr,a dxn dz md the integrals of this system being of the form
ic general solution of the given equation will be
This conclusion would follow also from the principle laid down in Chap. xiv. Art. 2.
Linear partial differential equations in which the absolute term is wanting, and which are therefore of the form
xr dz . .^ dz_ Y dz _
5—2
68 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
may be termed homogeneous. As in this case one of the auxiliary equations is
dz = 0,
the general integral will be
ul} u^, ...... un_l "being found by the integration of the remain-
ing auxiliary equations
dx dx dx
When X^ JT2, ...... Xn do not contain z, the solution is best
exhibited in the form
6. Every linear partial differential equation can be converted into a homogeneous one containing one additional variable. For it is shewn in Art. 3, that if u = 0 be any integral of
x.7r + x*-^~- + x«r = x>
1 dx^ z dx2 dxn
then is.
v du , v du . v du , vdu
a homogeneous equation with a new variable.
From the general integral of this equation, that of the former one may be deduced by making u = 0. .
7. The solution of partial differential equations is some- times facilitated by introducing a new system of independent variables. The actual transformation is greatly facilitated by the following symbolical theorem.
THEOREM. If the partial differential equation
~~ dx
X — 4- X — 4- X ~~ — X
1 z ' ' "
ART. 7.] ADDITIONS TO CHAPTER XIV. 69
be expressed symbolically in the form
Az = X, in which
A_ Y ^ 4- T — \-Y d
A = JL -j -- r -A-3 T~^ • •• T -A* T — ,
1dxl dx2 dxn
then, if ylt y«, ...... yn be a new system of independent vari-
ables given in expression as functions of the old ones, the transformed equation will be
For, regarding z as a function of ylt yt, ...... ynt we have
dz_^dz_dy, + dz_dy* dz dyH
dx^ dy^ dxt dyt dxl ' ' dyn dxl '
dz dz dif dz dy, dz dv_
— •"! _j «y 2 i '?* .
, dxn dyl dxn dyt dxn" dyn dxn'
whence, substituting in the given equation we find, as the total coefficient of ~ , the expression
.
~H ••• T •*** ~i — » dxa * dxn '
or symbolically, Ayt ; and so on for the other coefficients. The result then is
dz .. . dz ,,
It remains only after calculation of Ayu Ay2, ...... AyB, as
functions of cc1? xa, ...... xn, to express these functions and JL
intermsofy^y,,. ..... yn.
[It appears from the manuscript that an example was to have been supplied here.]
70 ADDITIONS TO CHAPTER XIV. [dl. XXIV.
[The next Article may "be considered supplementary to Chap. XIV. Art. 10.]
Singular Solutions of partial Differential Equations.
8. Legendre's theory developed in Chap, xxill. for ordi- nary, may be applied also without essential change to partial, differential equations. Regarding the independent variable z as receiving an infinitesimal change Bz through infinitesimal change, not in the values of the independent variables
X-
but in the values of the arbitrary constants of the complete or in the forms of the arbitrary functions of the general inte- gral, and performing upon the given equation the operation denoted by B, we shall obtain a linear partial differential equation for determining the general value of Bz corresponding to any particular given value of z. If that linear equation be of a lower order than the differential equation given, then the equation expressing the value of z + Bz will be a limiting form of a solution less complete or less general than the com- plete or general solution of the differential equation given, and the given solution, formed by making the infinitesimal constants in the limiting form actually 0, will be singular.
Conversely, to deduce singular solutions without the know- ledge of the complete or the general integral, we ought to construct the equations of condition for the reduction of the equation determining Bz to a lower order than the equation given, and the most general solution of the differential equa- tions of condition so formed, will be the most general expres- sion for the singular solutions of the differential equation given.
Ex. (px -qyY<l + ±mx* (z — xp) = 0,
in which
dz dz
AKT. 8.] ADDITIONS TO CHAPTEE XIY. 71
Representing the first member of the equation by Fy we have, on operating by 8,
dF dSz dF d$z
-j- ^r + -j- -j- + -7- 6z = o, dp ax dq ay dz
and the conditions
dF dF
~ = 0, °-f- = 0, dp dq
necessary to reduce the equation for 8z to a lower order give (px -qy)q- 2mx3 = 0,
From these we find p =
definite and simultaneous values of p and q, which being sub- stituted in the given equation lead to
z =
and this, as it gives the same values of p and q as those obtained before, will necessarily satisfy the given equation. It is therefore a solution, and from the nature of the analysis, a singular one.
Legendre shews that this singular solution is also dedu- cible from the general integral of the given partial differen- tial equation. That integral is the result of the elimination of a from the two equations
(a) + az — mxy = 0, {</> (a) - ax} <f> (a) - 2x$ (a) + z = 0.
To deduce the singular solution he supposes <j> (a) to be not simply a function of a, but a function of a and of one or both of the independent variables. He expresses the varia-
72 ADDITIONS TO CHAPTER XIV. [CH. XXIV.
tion of <£ (a) derived from this new source by S, and operating on the first equation with 8, finds
(a) - 2ax} S<£ (a) = 0 ; therefore <j> (a) = ax'
Substituting this in the equations of the general integral, and eliminating a, we find
z — 2m*x%y* as before.
Legendre states his theory of the derivation of the singular solutions of partial differential equations from the equations themselves with great brevity, but still as a general theory. Arid there is nothing in the statement that carries with it any apparent restriction upon either the order or the degree of the equations given. Until however we are in possession of a perfect theory of the genesis of partial differential equations we shall not be entitled to say that Legendre's theory of their singular solutions is a perfect one; for until then we cannot even define, in a perfectly general way, the nature of the operation denoted by 6.
[The next three Chapters all relate to the subject of partial differential equations of the first order. The manuscripts do not appear to have received their final revision from Professor Boole. It is certain that he intended the contents of Chapter XXV. to form a part of the new edition ; and it is highly probable, although not certain, that the contents of Chapter xxvi. and Chapter xxvu. were also to be included.
The three Chapters are mainly derived from two memoirs by Professor Boole, published in the Philosophical Trans- actions.
The first memoir is entitled On Simultaneous Differential Equations of the First Order in which tlie Number of the Variables exceeds by more than one the lumber of the Equa- tions : it occupies pages 437... 454 of the Philosophical Trans- actions for 1862.
The second memoir is entitled On the Differential Equa- tions of Dynamics. A sequel to a Paper on Simultaneous Differential Equations: it occupies pages 485... 501 of the Philosophical Transactions for 1863.
The first memoir was finished before Professor Boole had seen Jacobi's researches, which are cited at the beginning of Chapter xxvi ; these researches indeed could only just have been published. In his second memoir Professor Boole describes Jacobi's methods, refers to his own already pub- lished, and points out the nature of the connexion between them.]
( 74 ) [CH. xxv.
CHAPTER XXV.
ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFEREN- TIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSO- CIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS.
1. THE term simultaneous is here applied to a system of partial differential equations, to signify that in that system there is but one dependent variable, the general expression of which, as a function of the independent variables satisfy- ing all the equations at once, is the object of search. All linear partial differential equations of the first order being re- ducible to the homogeneous form, we shall presuppose this reduction here. Under this form indeed the problem actually presents itself in Geometry, in the theory of partial differential equations of the second order, and in Theoretical Dynamics.
We are sometimes led, in connexion with the same class of inquiries, to systems of ordinary differential equations marked by the peculiarity that the number of the variables exceeds by more than one the number of the equations. Such systems are intimately connected with the former — stand to them indeed in a similar relation to that which the Lagrangean auxiliary system bears to the single partial dif- ferential equation from which it arises. The theory which explains this connexion, and grounds upon it the method of solution of both systems will form the subject of the present Chapter.
Connexion of the Systems.
2. PROP. I. The solution of a system of simultaneous linear partial differential equations of the first order may be
ART. 2.] LINEAR PARTIAL DIFFERENTIAL EQUATIONS. 75
made to depend upon that of a system of ordinary differential equations of the tirst order in which the number of the vari- ables exceeds by more than one the number of the equations.
The system of partial differential equations being reduced to the homogeneous form, Chap. xxiv. Art. 6, let n be the
number of the equations, a^, a*8, xn^r the independent
variables, and P the dependent variable.
Then from the n given equations determining
dP dP dP_
dx^ dx^ ' " dxm*
we obtain an equivalent system of equations which, by trans- position of its terms to one side, assumes the reduced form
dP dP dP dP
-r- + An-r — + ^-7 — +^,r 7 = 0
dP dP dP dP
- — H-421-i l-^22~/ — + ^T/ — ~ \. n\
™ + Ang-+A,g- + A,~ = 0j
Multiplying these equations by the arbitrary constants
respectively, and adding the results, we have
X — \ — X —
1 dx. * dx. ' " dxn
LA '•
(2),
76 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
a single partial differential equation which, on account of the
arbitrariness of X,,X2, Xn, is equivalent to the system
from which it was formed.
Of this equation the Lagrangean auxiliary system will be dxl _ dxa _ dxn
fef^TM: W-
whence, eliminating X,,X2, Xn, we have the system of
ordinary differential equations
- Andxl - A2idx2 ....-Anldxn =
.... -Andx = 0
dx^r - Airdx, - A,rdxs . . . .-Anrdxn = 0
These equations being included in the previous system (3), any integrals
u = a, v = b, w = c, &c.
of them will be integrals of it. Therefore u, v, w,... will be values of P satisfying the partial differential equation (2). For they will be the only values which can satisfy it inde- pendently of Xj, X2, ...... Xn. Hence they will satisfy the
equivalent system (1), and the general integral of that system will be
F(u, v, w>,...)=0 ................. (5),
the form of F being arbitrary.
Thus the relation of the system (4) to the system (1) is the same as the relation of the auxiliary system oif a single linear
ART. 3.] EQUATIONS OF THE FIRST ORDER. 77
partial differential equation to that equation. And the ground of this relation is seen to be the same in both cases. The one form necessitates the other.
3. Instead of employing the above mode of deducing the auxiliary system, we might employ the following which is practically more convenient.
Since any value P which satisfies the partial differential equations determines P=c as an integral of the ordinary sys- tem, the latter must be consistent with tZP=0 in its de- veloped form
dP,dP dP ,
„,. . ^. dP dP dP
5V E- £„
by means of the n given equations (1), we have dP
(axn+l A.lldx1 M.21dx.2 j3.niaxn)
dP
J^T~ (dx^r - Alrdxt - A^dx.2 ...... - Anrdxn) = 0.
Whence, equating to 0 the respective coefficients of
dP dP dP
dx^ dx^' " dx^
we have the system (4).
In the same way we can pass from the system of ordi- nary to that of partial differential equations. From the, equa- tion dP = 0, in its developed form, we must eliminate a number
78 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
of differentials dxlt dx^... equal to that of the given equa- tions, and then equate to 0 the coefficients of the remaining
differentials.
4. Lastly, the formal connexion of the two systems should be noticed. The partial differential equations "being given in the reduced form (1), the ordinary system may be constructed as
follows : For any differential coefficient, as -^ — ,- in any
_ cfen+i ^
column after the first, write the corresponding differential c7.rn+1, subtract from this the sum of dxv dxz, dxn, mul- tiplied respectively by the descending coefficients of that column, and equate the result to 0. The system of equations thus successively formed will be the auxiliary system sought.
The transition from the ordinary to the partial system may be effected by the same rule, substituting only differentials for differential coefficients.
[It appears from the manuscript that an example was to have been supplied here.]
Up to this point the theory of systems of partial differen- tial equations is in analogy with that of single equations. But here a difference arises. We do not know beforehand what number of integrals a system of ordinary differential equations, in which the number of variables exceeds by more than one the number of the equations, admits.
The theory which removes this difficulty will be developed in the following sections. It will be shewn that a system of linear partial differential equations which admits of solution by the assigning to the dependent variable a value which satisfies all the equations in common, must either itself satisfy a certain condition, or be capable of being developed into a new but equivalent system which will satisfy that condition. It will be shewn that when that condition is satisfied, the auxiliary system of ordinary, is capable of expression as a system of exact differential equations determining the inte- grals sought.
ART. 5.] EQUATIONS OF THE FIRST ORDER. 79
It -will be found convenient to express by a single symbol the aggregate of the operations to which the dependent vari- able is subject in the expression of a partial differential equa- tion. Thus the equation
dz dz dz
di+xd*+ydy=Q
may be expressed in the form
if we assume
= —- —
dt dx
A = —4- —4- —
'
Under this convention the following proposition is to be
understood.
5. PROP. II. If AP = 0, A'P=0 represent any two homogeneous linear partial differential equations of the first order, then will
(AA'-A'A)P=0
also be a homogeneous linear partial differential equation of the first order, and it will be satisfied by all the common integrals of the equations from which it is derived.
First, the equation will be linear. For, let x, y represent any two variables whatever, or the same variable repeated, out of the set xl — xn, and let A, B represent any functions of the variables xl xn. Then A may be represented by a
series of terms of the form A -r- , and A' by a series of terms
ax J
of the form B -j- . Hence (AAr — A' A) P can be expressed by a series of terms of the form
, d
80 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
which, on effecting the differentiations, becomes
A—^-Bd~ ~ dx dy dy dx '
the terras containing the second differential coefficients of P mutually destroying each other. Hence the equation
(AA'-A'A)P=0
will "be a homogeneous linear partial differential equation of the first order.
The constitution of the coefficients of this equation is easily determined. For suppose the given equations to be
dP . dP . dP _
-a-, ~~, ~r -"•» ~i ...... «" -"n ~j — V)
1 cfcC 2 dx dx
, d A»T> A = £-...... + ^, A-
then the equation
(AA'-A'A)P=0
may be written in the form
and, since terms involving second differential coefficients of P will disappear, this becomes
ART. 6.] EQUATIONS OF THE FIRST ORDER. 81
We see from this that the form of the, result is the same as if the A or A' from either equation operated only on the coeffi- cients in the other equation.
Secondly, the above equation will be satisfied by all the common integrals of the equations from which it is derived.
For, let <£ = c be a common integral of
AP=0 and AT =0, then
Performing on these the respective operations A' and A, operations which involve only differentiation together with algebraic processes, we have
A'A<£ = 0, AA'<£ = 0,
whence, by subtraction,
AA'<£ - A'A<£ = 0, or (AA' - A'A) <j> = 0,
from which it appears that (f> is also an integral of the equation
(AA'-A'A)P=0,
as was to be shewn.
6. PROP. III. If by the above processes of reduction and derivation we convert a system of partial differential equa- tions into a new system, such that if expressed in the form
A1P=0, A2P=0, ...... AmP=0,
the condition
shall for each pair of equations be identically satisfied, then the system of ordinary differential equations corresponding to this new system will admit of reduction to the form of exact differential equations, the integration of which will enable us to construct the general value of P satisfying the system given.
R. D. E. II. 6
82 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
1st. Suppose the given system of n equations reduced to the form (1), marked by the peculiarity that n of the differen- tial coefficients appear only in successive equations and with the coefficient unity. Then taking any two of those equa- tions (we select the first two), we have
A ^ A ^ A ^
l~ dx* ud^~ ...... + lrd^~'
ajc\ aXn+i axn+r
from the forms of which we see that the derived equation
(AA-AA)J> = o
cannot contain either
— or —
dxi dx%
It can only, as appears from Art. 5, contain the differential coefficients
dP dP
and must be of the form
dP dP dP
~ + ~
It cannot therefore be an algebraic consequence of any of the equations of the system (1) from which it was derived. It is, unless by the vanishing of JSl, ....Br it present itself as an identity, a new equation algebraically independent. Com- bining this with the former ones, we have a system of n + 1 equations admitting of the same reduction as to form fol- lowed by the same subsequent process of derivation. And the result of each of these completed steps is to convert the system into one containing one equation more than before ; but containing in each of its equations one term fewer than before. The process must then end either in the genesis of a system of partial differential equations such that the further
ART. 6.] EQUATIONS OF THE FIRST ORDER. 83
application of the process of derivation of Prop. IT. shall only lead to identities, or in the emerging of the system
-
-; — — U,
dx
The latter supposition would imply that P is a constant. The consequences of the former we proceed to examine.
The final system of linear partial differential equations will be of the same type (1) as the original system, but will differ from that system in that n will be increased, and r diminished by the same amount. We shall therefore simply state the form (1), only under the condition
and with the altered values of n and r.
First, then, the common integrals of the new system will be the same as those of the original system. This is evident from Prop. II.
Secondly. If we write
m = n + r,
the first equation of the system (1) will be
and the auxiliary Lagrangean system of this will have m — 1 independent integrals
among which the n — I known integrals (Chap. XXIY. Art. 5)
*2=C2> *3 = C3> Xn = CH
6—2
84 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
are included. And the general value of P satisfying the above first equation will be
P=F(ul,u3....um_1}.
The assumption P=x1 would not satisfy the said equation, for it would lead, on substitution, to 1=0. Hence we infer
that while the functions MI} u2, wm_, are independent with
respect to each other, they are also independent with respect to ajj, so that the m functions ult u2, wm_1, xl, are mu- tually independent in the sense explained in Chap. xxiv.
Let us now transform the equations of the system (1) after the first by introducing ult u2, ..... um_:, xt as independent variables. Those equations being
A2P=0, ...... AnP=0,
the result of the transformation will be (Chap. xxiv. Art. 7) dP .dP . dP .. .dP
ar
dP .dP .. . dP .. .dP
But P = xl being an integral of each of the equations of the system (1) except the first, as appears from their forms, we have
thus the last terms in the transformed system vanish. Further, the coefficients of the remaining terms reduce to functions of MI} w8, ...... wm-1 merely. For, considering the coefficient A2w,,
we have
ART. 6.] EQUATIONS OF THE FIRST ORDER. 85
which, since ^lul = 0 reduces to
Hence &sul must be a solution of AjP= 0, and therefore a function of «,, ut, ...... um_1. And so for the others. It
results therefore that the transformed system is
.dP .. .dP . dP
«,, ua, ..... um_1 being the actual independent variables of the system.
But the transformation having involved no loss of gene- rality, for a new system of ra independent variables was simply substituted for an old one, the condition
satisfied before, will continue to be satisfied in the new sys- tem represented symbolically in the form
A,P=0, ASP=0, ...... ABP=0.
Any common integrals of this system will also be common integrals of the previous system. For as functions of
they will satisfy the first equation of that system, and they will satisfy the other equations, because the present system is but a transformation of those. The converse is equally mani- fest
Thus a system of n partial differential equations contain- ing m independent variables and satisfying a certain condi-
86 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
tion, has in virtue of that condition been converted into a system of n — 1 equations between m — 1 independent vari- ables, and satisfying the same condition. This then is con- vertible into a similarly constituted system of n — 2 equations containing m — 2 independent variables, and so on till we arrive at a final single partial differential equation containing m — n + 1 independent variables. This equation has m — n, that is, r integrals, and these are the common integrals of the system (1).
But the system of ordinary differential equations corre- sponding to (1) is in number r, and is satisfied by all the common integrals of that system. Hence these differential equations must admit of reduction to the exact form.
7. We may deduce from the above investigation the fol- lowing Rule.
To integrate a system of simultaneous linear partial diffe- rential equations of the first order.
EULE. Reduce the equations to the homogeneous form (1), express the result symbolically by
A,P = 0, A2P=0, ...... AnP = 0,
and examine whether the condition
is identically satisfied for every pair of equations of the sys- tem. If it be so, the equations of the auxiliary system, Prop. I., will be reducible to the exact form, and their inte- grals being
u = a, v = b, w = c, ......
the complete value of P will be F(u, v, w, ...), the form of F being arbitrary.
If the condition be not identically satisfied, its application will give rise to one or more new partial differential equa- tions. Combine any one of these with the previous reduced
ART. 7.] EQUATIONS OF THE FIRST ORDER. 87
system, and again reduce in the same way. With the new reduced system proceed as before, and continue this method of reduction and derivation until either a system of partial differential equations arises between every two of which the above condition is identically satisfied, or, which is the only possible alternative, the system •
dP dP
j-=0> -r- = 0, ... dx^ dxa
appears. In the former case the system of ordinary equations corresponding to the final system of partial differential equa- tions will admit of reduction to the exact form, and the gene- ral value of P will emerge from their integrals as above. In the latter case the given system can only be satisfied by sup- posing P a constant.
Ultimately then the determination of P depends on the solution of a system of ordinary differential equations reduci- ble to* the exact form. This does not mean that each equation of the system is reducible to the exact form, but that the equations may be combined together so as to form an equal number of equivalent equations of the exact form. Generally when we know this combination to be possible it is easy to effect it, and best to endeavour to do so. We might how- ever employ the method of the variation of parameters as fol- lows. Supposing p the number of differential equations make all but p + 1 of the variables constant, integrate the reduced system, and then seek to satisfy the unreduced system by the same series of integrals with the arbitrary constants as new variables. The successive integrations and transformations of this method would amount to the same thing as those upon which the second part of the demonstration of Prop. in. rests*.
Lastly, given a system of ordinary differential equations containing a superfluous number of variables without know- ing how many integrals they admit, we must, supposing P = c to be any integral, construct the corresponding system
* It was thus indeed that the author was first led to that theory.
88 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.
of homogeneous partial differential equations satisfied by P, and apply to them the foregoing Rule.
8. Ex. Required the integrals of the simultaneous par- tial differential equations
dP .dP . dP
_ + (a^ + y_^)_+(^_y)_=().
Representing these in the form A,P=0, A2P = 0, it will be found that the equation
becomes, after rejecting an algebraic factor,
3P.*P_
X dz + dt ~
and the three equations prepared in the manner explained in the Rule will be found to be
dP
TZ=^
dP
T. -r- = 0. at dz
No other equations are derivable from these. We conclude that there is but one final integral.
To obtain it, eliminate
dP dP dP
dx' ~dy ' dt
ART. 8.] EQUATIONS OF THE FIRST ORDER.
from the above system combined with
dP , dP, dP, dP, n -T-dx+-j-dy + -:rdz + --r-dt = (), dx dy ' dz at
dP and equate to 0 the coefficient of -y- in the result We
find
dz-(t + 3x*) dx - ydy - xdt = 0,
the integral of which is
An arbitrary function of the first member of this equa- tion is the general value of P.
[It appears from the manuscript that another example was to have been added here.]
( 90 ) [CH. XXVI.
CHAPTER XXVI.
HOMOGENEOUS SYSTEMS OF LINEAR PAETIAL DIFFERENTIAL EQUATIONS.'
1. THE theory of homogeneous systems of linear partial differential equations in which when expressed in the sym- bolic form
A1P=0, A2P = 0, AmP = 0 (1),
the condition
(AA-AA)^=0 (2)
is for all combinations represented by i and j satisfied in virtue of the constitution of the symbols Af, A;-, forms the subject of important researches by Jacobi (Nova Methodus... Crelle's Journal, Vol. LX. p. 1). The following are the most important of his results.
1st. An integral of any one equation of the system being found, other integrals of the same system may be obtained without integration, by a process of derivation founded upon the condition (2).
Let <j> be an integral of the first equation of the system. Then is the equation
A^ = 0 identically satisfied.
Also the condition (2) being satisfied in virtue of the con- stitution of the symbols, we have
ART. 1.] HOMOGENEOUS SYSTEMS &C. 91
and in particular, making i=l, and separating the terms,
which reduces by a prior equation to
It appears from this that A/£, if it do not reduce to a con- stant, is an integral of the first equation A^ = 0, and, if it prove to be not a mere function of <f>, a new integral.
This process may be repeated upon the new integral with a similar alternation of results. It will be evident from this that if we confine our attention to the two equations
A1P=0, A2P=0,
and suppose, as before, <j> to be an integral of the first, then will
or, as these may be expressed,
be also integrals of the first equation; and this process of derivation may be continued until we arrive at an integral A/c^ which is not independent, but is expressible as a func- tion of prior integrals
and, sooner or later, such a result must present itself, since the number of independent integrals is finite.
It is further seen that the most general symbolic form of an interal derivable from the root interal <> is
a, /?, ...... ft, being positive integers.
The above remarkable theorem was in some degree antici- pated by the researches of Poisson.
92 HOMOGENEOUS SYSTEMS OF LINEAR [CH. XXVI.
2ndly. Jacob! shews how by the aid of such derived in- tegrals of the first equation of the system a common integral of the first and second equation may be found, and how from this integral and its derived series a common integral of the first three equations of the system may be found, and so on, until a common integral of the entire system has been as it were built up out of previous integrals of less general appli- cation.
Let <£, <£', <£", ...... ^-^ represent a series of independent
integrals of the equation A1P = 0, of which </> is the root in- tegral, and the rest are derived from it by successive applica- tions of the operation denoted by A2 , so that
also let A/<£ be not a new integral but a function of
Now <f>, <£', ...... (f>^~^ being particular integrals of AjP=0,
the function F(<f>,<j>', ...... ^~1') will also be an integral ^ of
the same equation irrespectively of its form. Let us inquire whether the form of the function can be so determined as to render it also an integral of the second equation AaP= 0.
We have then to satisfy the equation
By the principles of the Differential Calculus this equation assumes the form
But
lastly, A.,^-1* may by hypothesis be expressed in the form ', ...... ^-a)). Thus the equation to be satisfied is
ART. 1.] PARTIAL DIFFERENTIAL EQUATIONS. 93
dF
a linear partial differential equation of which the auxiliary system is
..
-»
Now the integration of this system may be made to depend upon that of an ordinary differential equation of the (p — I)* degree between the two variables <f>:*~1} and <j>.
For we have
d<f> $
Differentiating the last equation with respect to <f>, and attend-
di<f><i'-V .
ing to the former ones, we shall be able to express J in
terms of the variables <f>, <f>, ...... ^-1). Proceeding with
this in the same way and continuing the process we shall be able to express the series of differential coefficients
'
in terms of <£, <f>, ..... ty*' eliminating <£', ^>", ..... ^>'M~^, we between
Lt Y/
From these p, — 1 equations, 'G shall have a final equation
' d<j> '" dp-L '
that is, a differential equation of the (jj, — 1)°* order between and $<»-».
94 HOMOGENEOUS SYSTEMS OF LINEAR |_CH.
The complete integral of this equation will be of the form
Differentiating this //, — 2 times in succession with respect to <b, and continually substituting for the differential coefficients of (^"^ their values as before assigned in terms of
we shall have a system of /* — 1 equations connecting the above variables with the constants c1? c2, ..... cM_i. Finally, solving these equations with respect to the constants, we shall possess the integrals required in the form
and each of these will be a common integral of the first two equations of the given system (1).
[On the back of a page of the manuscript the following paragraph occurs, which seems to have been intended as a simplification of the preceding argument which begins with "The complete integral."]
Suppose that a first integral of the equation can be found. Its form will be
Substitute in this for the differential coefficients of <f>(t*-l> their values before assigned in terms of </>, </>', (f>",...(f>lp~l\ and we have an integral of the system (3), and therefore a com- mon integral of the first two equations of the system (1).
[We now return to the place at which we inserted a para- graph.]
Just in the same way Jacobi deduces a common integral of the first three equations of the system (1). For representing
ART. 2.] PARTIAL DIFFERENTIAL EQUATIONS. 95
any one of the first members of the above system by i/r. and deriving thence the new independent integrals Asi/r, A^^r,... he substitutes an arbitrary function of these for P in the equation
ASP=0.
It is evident that the solution of the partial differential equation so found will again be reducible to that of an ordinary differential equation between two variables. And so the process is carried on till all the equations are satis- fied.
2. The above remarkable process was developed by Jacob! in connexion with the theory of non-linear partial differential equations of the first order. In that particular connexion it admits of certain reductions tending to diminish the order of the differential equations to be integrated. But these do not affect the general principle of the method. It was in this special form that the theory of the solution of simultaneous linear partial differential equations originated. Jacobi does not consider the theory of equations in which the condition (2) is not satisfied ; but the language in which he refers to the condition shews that he had speculated upon the general problem — and it is difficult to conceive that he should have meditated upon it and not arrived at its complete solution.
[The manuscript here gives the first two words of the passage from Jacobi's memoir which is quoted in the Philosophical Transactions for 1863, page 486.]
(96 ) [CH. xxvn.
CHAPTER XXVII.
OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER.
1. IN treating the present subject we shall first consider that class of non-linear partial differential equations of the first order which involves two independent variables, and then proceed to the general theory. The reason for this procedure is that the particular theory, though of course in- cluded in the general one, rests upon a somewhat simpler basis, and it was in fact developed by the labours of Lagrange and Charpit long before the general theory was known. The latter we owe to the independent researches of Cauchy and Jacobi.
[Here the manuscript refers to the matter contained in Chap. xiv. Arts. 7 to 12 inclusive; and then passes on to the general theory.]
General Theory. 2. Given an equation of the form
the number of arbitrary constants a,, a2, ... an involved being equal to the number of the independent variables xv xz, ... XM we obtain by differentiation and elimination of the constants a partial differential equation of the first order. Of this the proposed equation is said to constitute a complete primitive.
ART. 2.] XOX-LINEAR PARTIAL DIFF. EQUATIONS &C. 97
The form of the above process which it seems best, as throwing light upon the inverse problem of deducing the complete primitive from the partial differential equation, to employ, is the following. Let the given primitive, solved witli respect to one of the arbitrary constants av be presented in the form
f(xlt ... xn, z, oa> ... aj =a^ ...... (1).
Differentiating with respect to each of the independent vari- ables we have a system of n equations of the forms
(2).
These n equations enable us first to eliminate the n — 1 constants a^, ...... an, and so deduce the partial differential
equation sought in the form
^fo,... x*,z,2>l,...jpj=0 ............ (3);
secondly to determine the n — 1 constants as functions of xlt ... #n, z,p^, ... pn in the forms
(xlt ... xMe,plt ... /)») =aa]
...........................
(xlt ... XH, z,plt ...pn)=anj
As the system formed of these n — 1 equations, together with the previous one, is merely another form of the system (2) obtained by directly differentiating the primitive, it follows that if from these equations we deduce the values of^, ...^m as functions of xlt ...xH,a3, ...an, and substitute them in the equation
they will render that equation integrable, and its- integral will be the complete primitive (1), the constant a^ being re- gained by integration.
B.D.E. IT. 7
98 NON -LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
Examining the system (3), (4) we see that the first mem- bers of all the equations which it contains are functions of xl, ... xn, 2, jp1? ... pn, while the second members are con- stants. The question then arises, What mutual connexion exists among these functions in virtue of which they yield values of plt ...pn, which render the equation (5) inte- grable?
The answer to this question must involve the entire theory of the solution of partial differential equations of the first order, so far as relates to the determination of a complete primitive. Given a partial differential equation of the form (3) it is evident that if we can construct a system of associated equations (4) possessing the character above described, the final value of z obtained by integration of (5) will both satisfy the given equation and contain the requisite number of arbitrary constants. It does not follow from this that it will be the only complete primitive, but it will be a complete primitive.
3. The relation sought is expressed in the following Proposition :
PROPOSITION. If
<£(#,, ... xn, ztplt ...pn)=b
represent any two out of a system of n independent equations such that the values ofp^ ...pn, thence determined would make the equation
dz =d
integrable, then the first members of these equations lei/iff represented for simplicity by F and 3>, the condition
F\d$> dF/d®
the summation extending to all values of i, from 1 to n inclusive, will be satisfied identically.
ART.
EQUATIONS OF THE FIRST ORDEE.
99
Reciprocally, if the above condition be satisfied identically for each binary combination of functions in the proposed system of equations, and if these functions be independent, then the values ofpl .... p^as functions ofx^ ...xn, z, which they yield, tcill make the equation
dz =
integrdble.
It will be convenient to begin with the particular case in which the proposed equations do not explicitly contain z, the particular pair to be considered being represented by
Differentiating with respect to ar£, and regarding^, ...pn as functions of the independent variables, we have
dxi
d® ^i+ +^^=0 I
(6),
which we may give the form
*E - - 1 — dp/' dxt ' dp* dx\
d® _ ^ d® dp}-
(7),
ie summation with respect toy extending fromy= 1 to j = n iclusive.
From the first of equations (7) multiplied by -5- subtract
7—2
100 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
- rJW the second multiplied by -j— , and sum the result with respect
to i from i= 1 to i = n inclusive. We have
2 ldF_ d<& _d_F d3>\ "l \f[xt dpi dpi dxj
___ ( ,
'* 3 \dpj dpi dxi dpi dpj dx
The expression under the double sign of summation in the second member vanishes when *=,/; we may therefore re- strict the summation to unequal values of i and j. Now as for any particular combination of values, e.g. 2, 3, there would exist in the completed member both the terms cor- responding to t = 2,J = 3, and those corresponding tot/ = 2, i=3, it is evident that if we employ the symbol 2y to denote summation with respect to different combinations of i andj, the second member of the last equation may be expressed in the form
^ fdF d<£> dpj _ dF d3> dp,. v \dj)j dpt dxi dpt dpj dxi
dF^d^dpi_dFd^ dpi dpj dxj dp3- dpi
^ (fdF d$> dF d®\ fdpi dp,
Q-M > J / _ _ . _ _ j I _-*. mm _ _*_JL
lj \\dpi dp; dpj dpi/^dXj dx
so that the equation (8) becomes
dF^ dpi
_^ (fdJ^d^_Wd^\(d^_dpJ\\ ^ \\dfr dPi dPj dpj \dx}- dxj}"
The number of terms of which the second member ex-
M ( ft ^_- 1 j
presses the sum is thus - - , and it will be observed that
ART. 3.] EQUATIONS OF THE FIRST ORDER. 101
as to any particular term it makes no difference in what order the numerical values of i and j are assigned to these quan- tities; e.g. whether for the combination 2, 3 we make t = 2, j = 3, or i = 3, j = 2; but we must confine ourselves to one order.
Now when the equation dz =
is integrable in the manner here supposed, we have for all combinations of i andj,
All the terms in the second member of (9) therefore vanish, and we have
2 fdF_ d®_d]^ d®\ "* \dxi dpi dpi dxj
This is the direct form of the Proposition under the parti- cular limitation supposed.
As F, 3> represent, under the same limitation, any two of the first members of the n equations (3), (4), which determine
pl} '-.pn, there will exist —~- — - equations like the above.
21
It is usual to employ for brevity the notation
2 (dF_ d^_dF d^\_ i(dxi dp, dp, dxj~l j>
and this being done the above system of equations expresses the - - - functions of the form [-^^ as linear homogeneous
s S .1 n (n ~ 1) , «, • j? -L _e dpi dp,
functions of the - - quantities of the form -~- — -/- .
& dXj dx\
It is hence that the vanishing of the latter series of quantities secures the vanishing of the former.
The converse truth will therefore be established by shewing that the — ^— — - quantities of the form -~ — -f1 are. when
102
NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
Flt Fz , . . . Fn are independent with respect to p^ p2, ...pn, expressible as linear homogeneous functions of the — =-?
9
functions
To avoid complexity of expression I shall establish this for the particular case of n = 3, and shall shew that the reasoning is general.
The functions Flt F^ F3, being independent with respect z the determinant
df
dF
<*&'
does not vanish. This determinant we shall denote by A.
In (9) writing for .Pand <3> first F2 and F3, secondly F3 and Flt thirdly Ft and Fa, we have on changing signs the system
(10).
dp 3 dp2J \dxa dxf^ dp3 dpz ~dpj \dxs dxj
Multiply the first equation by -j— • , the second by -, 2, the
third by -j- - and add. Then dp,
dp,
dp, *'
ART. 3.] EQUATIONS OF THE FIRST OBDEE. 103
whence as A does not vanish we have, on dividing by it, the
function / -- — ~ expressed as a linear homogeneous function dxs dxz
of [F^, [F.F'l, and
In like manner multiply ing the equations by -^ , -=— - , -y-1
J dPi dp% dp,
respectively, and dividing by A, we obtain -j- — -^-l as a
CLX - d3ts~
similar linear homogeneous function, and lastly, multiplying
dF c!F
, by
-7— -, -j—,
, and proceeding as before, we obtain
jp — -?* as a similar linear homogeneous function. dxa dx1
From all which it follows that when [J^FJ, [F3FJ, vanish, then
_ fa dj>^_ dp^
dx.
dx.
will vanish also.
The reasoning is general in its nature. If Ft. F2, ...F are independent with regard topltptt ...pn, the deteriuiuant
= A
loes not vanish. This determinant is from its constitution a determinant linear and homogeneous, not only with spect to any row or column of elements, but also with ispect to the possible binary combinations which can be formed of two rows or columns, ternary out of three rows or )lumns, &c. provided that these combinations are themselves }f the form of determinants. In the language of the theory uch combinations are called minor determinants. Hence if re construct the system of equations represented by (10), and
104 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
observe that the coefficients of any particular term of the form
•*-*• — ^ in the several equations form a system of such dxj dxi
minors to the general determinant (11), it will be plain that the equations can by multiplication and addition be brought to a form in which the coefficient of that particular term will be A. At the same time the coefficients of all the other
terms of the form -~ — §p will vanish. For a little atten- dXj dXi
tion will shew that they will be what the determinant A would become on making two of its columns or rows of ele- ments equal, and therefore will be identically equal to 0.
Thus the Proposition is generally established for the case in which z does not explicitly appear in the functions
F F F -*• i j * g i ..... -*• » •
When z does appear in those functions the equations (6) will be replaced by
. ., ..
dxt "* dz dpt dxi dpn dxj,
d<£> d<& J<E> dpl d<& dpn _
dxi ** dz dpl dxi dpn dx^
from which it is seen that the theorem above established will only need to be changed into the form employed in the state- ment of the general Proposition.
As the above is one of the most important propositions in the entire theory of Differential Equations, it may be desire- able to illustrate it by examples.
[There are no examples in the manuscript.]
4. We resume the general theory.
The integration of non-linear partial differential equations may be effected by two distinct methods, both resting upon
ART. 4.] EQUATIONS OF THE FIRST ORDER. 105
the ground of the above Proposition. The first of these methods, originally established by a different analysis from that which will here be employed, was discovered by Cauchy (Exercices d' Analyse), and rediscovered by Jacobi (Crelles Journal). The second method, discovered by Jacobi at a later period, forms the subject of his posthumous memoir, Nova Methodus...
Cauchy s Method.
We will, as before, begin with the case in which z does not appear explicitly in the proposed partial differential equa- tion, which we shall represent in the form
•Fi(*i> ..... *., plt ..... lO =0 .............. (1).
We have seen that to find a complete primitive of the equation it is necessary and sufficient to construct a series of equations
F*(xv ..... *-» Pi. ..... /0=«tl
............................... ............ (2),
F»(*i ..... ar.»l»i» ..... |0=*«J
such that not only shall the conditions
connecting the new functions F3, ..... FH with Flt be identi- cally satisfied, but also the series of conditions
GO,
Fa and Ft representing any two of the new functions re- ferred to.
The first of the above series of conditions amounts to this, that jP2, ..... Fn must be integrals of the partial differen- tial equation
o ......................... (5).
106 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
It is the peculiar aim of Cauchy's method to determine the integrals so as to cause the second series of conditions to be satisfied also. And it is shewn that this will be attained if the integrals of (5), which form the first members of (2), are such that the particular values which p^ ...._pn_1 assume when xn is made to receive any constant value, as 0, are differential
coefficients with respect to a^, xn^ of any single function
of those variables, the form of which may be arbitrarily assigned.
The necessity of this condition is obvious. If the general
values of p^ pn are differential coefficients of a function z
with respect to xlt xn, then the particular forms which
j? , p i assume when xn receives any constant value are
simply differential coefficients with respect to x, xn_1 of
what, z becomes under the same circumstances. To prove its sufficiency we must shew that when it is satisfied the condi- tions represented by (4) will be satisfied also.
Since Fa and Fb are integrals of [FJP] = 0,
= 0.. ,..(6).
Also, since if in (t) and (2) we give to xn a particular con stant value, as 0, and then in (2) regard pn as a function of
determined by (1), the system (2) will virtually contain only
X\1 ..... Xn~U Pl> ..... Pn-\1
of which jt?j, ..... pn_1 are differential coefficients of a single function with respect to x^ ..... xn_^ it follows from the pro- position of Art. 3, that any two functions Fa and Fb will satisfy mutually the condition
2i«.-i (dF* d]\ _ dFa dF\\ =. Q i-1 \dxi dpi dpi dxj
the differentiations having reference to
ART. -4.] EQUATIONS OF THE FIRST ORDER. 107
explicitly as they appear in Fa and Fb, and implicitly as involved in p». Thus the developed form of the above equa- tion is
i=n_! (/dF_a d_F» dfa ftF, dFb dp\ iU* + dpn dxj (<fr + dpn dpj
_ , (dp, + dpn dpj (dxt + dpu dx ~
the forms of ~ and -f^ beinor determined from (1).
Performing the multiplications, the above equations will be reduced to the form
_ dpi
_ dpn \dxt dpi dpi
dj\ ^=n_, (dpn dFa dpn dF«\ _ dpn2i=l (dx, dp, fa dxi)-
But from the form of the total differential of (1) we see that
df\ df\
dpn _ dxj dpa _ =~
Hence
dp,, dFa
i=.-i fPn d_Fa _ dp,, dFa\ (dxi dpi dp, dxj
(Wr 2f --i (dK dF. _ dj\ dFa\ \dpj : \dXi dpi dpi dxj
108 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
Now since by Art. 3
^=nfdF\dJ\_dF\dF\\=Q 1 1 \dxi dpi dp i dxj
we have
i dpi dpi dx
_
\dxn dpn dpn dx
therefore
_ t-1 \dxi dpi dpi dxj
= (^}~l (dl± ^ _ ^5 d_
\dpj \dxn dpn dpn dx In the same way
^i-.-i/^ ^_dP^ "~t~1 \clxi dpi dpi dx
_ \dpj \dxn dpn dpn dx
The substitution of these values in (7) gives
^n-i(dladF,_dF. dF>\ ~i=1 \dxi dpi dpi dxj
fdFY-1 (dF, fdf\ dFb _ dj\ dFb\ \dpj \dpn \dxn dpn dpn dxj
^_dF, fdF\ dF,>_dFt df].\\_0 <*P* \da* dpn dpn dxj] ~ '
ART. 4.] EQUATIONS OF THE FIRST ORDER. 109
\dxi dp i dpi dxj
_ =
dxn dpn dpn dxn
^n(dF,dF*_dF_adJ\\ = Zi=1U*i dfi dPi dxj
which is precisely the equation
We see therefore that to solve the partial differential equation
FI(*> ..... *„, P*, ..... jO=0»
it is only necessary to construct the linear partial differential equation
^n__
Zi=1Uri dpi dpi dxj~
and to obtain n — 1 independent integrals of this
Fn(x^ ..... xn, plt ..... ?„)=<*„,
such that if we determine from these conjoined with the given equation the values of pl,....pn, then those ofj91? ..... p shall, when xn is made constant, be the partial differential coefficients of one and the same function of xl} ..... x^ with respect to these variables in succession.
Now provided that we can find all the integrals of the above partial differential equation the particular determination required may be effected in the following manner.
The Lagrangean auxiliary system consists of 2n — 1 ordi- nary differential equations
dp* dx^ dxn
1.10 NON-LINEAK PARTIAL DIFFERENTIAL [CH. XXVII.
These admit of 2n — I integrals, one of which will be F1 = cl; and this will agree with the given equation if we make ct = 0. We have therefore, besides the particular integral Fv = 0,
2n — 2 integrals of the form
Now suppose it required to find a value of z as a function of xl , . . . xn , which shall satisfy the given partial differential equation, and shall reduce when xn = 0 (or any numerical con- stant.) to a particular given function of cc,, ...#„_!, which we will represent by •x/r (a^, ... a^J. Then on the assumption that xn = 0, we have first the given equation
z = Tjr(x1,...a:n_l) ..................... (10),
secondly the derived equations
_
_<ty (*,,... «?,_,)
(11).
Make in the 2n — 1 integrals xn = 0, and suppose at the same time o^, ... xn_l, p^ • ••JV,_i to receive therein the same values as in the above derived equations. Then from the 3n — 2 particular equations which we thus possess in the two systems united (particular because under the assumption that xn = 0), we can eliminate the 2n—l particular values of £Cj, ...#„_,, p ...pn_^ and so obtain n — I equations among the constants. These express the conditions which are necessary and sufficient in order that the values of pl , . . . pn^ thus derived from the integral equations may, when xn = 0, agree with the values assigned in (11). Accordingly if we substitute in these equations of condition for c2, . . .c2n_^ the general values </>2 , . . . fa^ we shall obtain n — 1 equations between xlt ... #„, pl, ... pn, which will at once be particular integrals of the system (8),
ART. 4.] EQUATIONS OF THE FIRST ORDER. Ill
and will possess the property that the values of pl , . ..pn^ which they in conjunction with Fl = 0 give will when xn = 0 reduce to the values given in (11). Hence these values with that of pn derived from the same equation will make
dz -Pl dx^ - ..... -pn dxn = 0
an exact differential equation. In the integral of this it will only remain to determine the constant so as to make the value of z agree with that given in (10). All the conditions will then be satisfied.
We may collect the results of the above investigation into the following Eule :
To obtain an expression for z as a function of the inde- pendent variables x^ ...xn, which shall satisfy the partial differential equation
and shall when xn is made equal to 0 (or to any numerical value) reduce to a given function of x1} ... x^_1} which we will represent by tyfa, ...o^J.
RULE. Construct the linear partial differential equation
^n_ =
Zi=l\dxi dp, dp, dxj
and forming its auxiliary Lagrangean system deduce its in- tegrals
<£2 = C2» • • • </>2»_l = C2n_l>
in addition to the known particular integral F= 0. Between the above integrals and the equations
eliminate, after making a\ = 0, the quantities
112 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
In the resulting n — 1 equations replace
C2b7 &> • "<W by </>,„_! ,
and we shall have a system of equations which with F=Q will determine values of^, ...^?n, which will render
dz -p1dxl - ...-pndxn
an exact differential. The integration of this will give the integral sought.
In the case in which the given partial differential equation is of the form
F(xlt ...xnt z, plt ...j?J = 0,
z being contained explicitly, the linear equation to be solved is dF dF\dP dFfdP
and the argument by which it is shewn that the integrals of this to be employed in conjunction with F=0 for the deter- mination of •pl , ... pn need only be so conditioned as to make p1, ...^n_i differential coefficients of one and the same function of #j , . . . aVi when xn = 0 is in character the same as that already developed in the present Article. It is only necessary
. . dF dF , dF to substitute in its exposition -, — \-pi -7- tor ^— , and so
for the other functions.
But as the auxiliary system
dx dxn dz
_dF~ _ dF_ _ dF dp, dp. &dPl Pndpn
~ dF dF dF^ dF"
dx^Pl dz dxn+Pndz
virtually includes the equation
dz -pidx^ - ... -pndxn = 0, the ultimate expression of the Rule will be as follows :
ART. 5.] EQUATIONS OF THE FIRST ORDER. 113
To obtain an expression for z as a function of a^, ... xn which shall satisfy the equation
... x
and shall when xn is made equal to 0 (or to any particular constant value) reduce to a given function ty (x1} ... a^) of the independent variables x1} ... x^.
RULE. Let
be the 2n — I integrals of the auxiliary system (12) which are additional to the particular integral F= 0. Make in these 2n equations arn = 0 and forming the further equations
z = -f (xlt ... or,.,), , ... ar,J
-- -
eliminate the 2n quantities a^, ... #„_!, z, plt ... pn. obtain n equations among the constants c^,...^.
Substitute in these equations <f>3 for c2, ... $,„ for Cj*, and we have n equations connecting #,, ... #„, z, _p17... ^?», from which with the aid of the given equation^, ... pn may be eliminated, and there will result a single equation connecting xl} ... o\ with z. This is the integral sought.
[It appears from the manuscript that an example was to have been supplied here.]
5. Cauchy's method is evidently a general one. But its generality is not of the same kind as that which belongs to Lagrange's solution of linear partial differential equations. It conducts us, not to a form embracing every possible
B.D. E. II. 8
114 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.
solution, but to a system of results from which every possible solution may be derived, by arbitrarily varying the form of the function which expresses the initial state of the dependent variable, that is the value of z when xn = 0, and then per- forming certain eliminations. To obtain a complete primi- tive we should only have to assume as the form of z when xn = Q a function of the variables xv ... xn_1 involving n independent constants. The form of this function is arbitrary. Each distinct determination of it under the conditions leacU to a distinct complete primitive. The number of such com- plete primitives is infinite.
There are some most important problems in which the knowledge of a single complete primitive is all that is re- quired. For this purpose the method of Jacobi which we shall now give may be employed.
Jacobi's Last Method.
6. Supposing z to be not explicitly involved in the given partial differential equation
which we shall as before represent by Fl = 0, the problem of the discovery of a complete primitive consists in the finding of n — 1 equations
F = o F =a
2 U2' ............ *• — Wn>
such that between any two functions FtFj the relation
0 ......... (1)
shall be identically satisfied. The values of p^ , ... pn deduced from the equations, by rendering
dz —pidxl — ... —pndxn= 0 integrable lead us to the complete primitive expressed by its
integral,
ART. 6.] EQUATIONS OF THE FIEST OEDEE. 115
Xow the idea upon which Jacobi's later methods rest is that of directly solving the different systems of linear partial differential equations flowing from the general condition (1), not of solving, as in Cauchy's method, one of those equations and then limiting that solution by conditions which virtually involve the satisfaction of the others.
It is evident that the entire series of — ^ -- - conditions
(1) will be satisfied if we determine Ft to satisfy the single equation
then F3 to satisfy the system of two simultaneous partial differential equations
then FI to satisfy the system of three simultaneous partial differential equations
and so on, until finally Fn is determined by the solution of the system of n — 1 partial differential equations
n] = 0, [F2FJ = 0, ...... [J^JPJ = 0.
Now all these are particular cases of the general problem of determining a function P which shall satisfy simultaneously the equations
[FtP] = 0, [FJP] = 0, ...... [FnP] = 0. (2)
F^ Fa, ... Fn being given functions between each pair of which the equation
is identically satisfied. Here P will represent in succession the series F,, F3, ... Fn.
The given system is one of homogeneous linear partial differential equations. It belongs to the class of systems the
8—2
116 NON-LINEAR PARTIAL DIFFERENTIAL [CH. xxvu.
general theory of which is discussed in Chap. xxvr. But it is not necessary to apply the theory in its general form. We need only a single integral ; for a single value of each of the functions jP2, F3, ... Fn suffices in combination with the given value of Z\ for the determination of a complete primitive. Now it may be shewn that the system is of the class dis- cussed in Chapter xxvi. If expressed symbolically in the form
A1P=0, A2P=0,... AnP = 0, the condition
(AA--AA)P=O,
will be identically satisfied. Hence Jacobi's method for the treatment of -systems of this kind may be applied.
That the system is of the kind asserted is a consequence of the following proposition.
PROPOSITION. If the equations
[«P]=0, [>P] = 0 are expressed in the symbolic form
AP=0, A'P=0, then the derived equation
(AA'-A'A)P=0 (3),
will be equivalent to
f[ uv] P | = 0.
^i=n d du d
For A = Zf=i K— -j -- -j— -j— ,
i dpi dpi dxj
. dp,, dpi dx '
ART. 6.] EQUATIONS OF THE FIRST ORDER. 117
Hence since - - is the coefficient of -y- in A'P, and -—
its coefficient in AP, its co efficient in the derivedequation (3; will be (Chap. xxv. Art, 5),
. dv A , du
A -j A -T-;
dxj ax.
or
or
d*u dv_ d*u \
t=n/du d*v du d*v _d
1 \dxi dpidxj dpi dxidxf dxi dp^dxj dp^
d ^,-=n fdu dv du dv
or
In like manner the coefficient of -^— is
Hence (A* - A'A) P = ^ (*& f -^ f )
•rl \ ^c, ^ dfi <faj'
whence the Proposition is established.
Applying this to the system (2) we see th'at any derived equation will be of the form
But [FtF~\ = 0 by the conditions given ; hence the condi- tion (AiAy — A;A4)P = 0, is identically satisfied.
The results of Chapter xxvi. being thus directly applicable to the system under consideration, we see that a common integral of the system (2) may be found by a series of alter-
118 NON-LINE AE PARTIAL DIFF. EQUATIONS. [CH. XXVII.
nate processes of integration and derivation. We begin by seeking an integral of the first partial differential equation. By a process of derivation, always possible, followed by the integration of a differential equation between two variables, we arrive at a common integral of the first two partial diffe- rential equations. Again, by a process of derivation followed by the solution of a differential equation we obtain a common integral of the first three partial differential equations. And so on, until a common integral of all is obtained.
7. Another solution of the above problem has recently been given. Beginning as in Jacobi's method by finding an integral of the first partial differential equation, a process of derivation agreeing in principle with Jacobi's, only more extended, may lead us without further integration to a point at which the discovery of a common integral of the entire system will depend only upon the solution of a single diffe- rential equation of the first order susceptible of being made integrable by a factor. Failing this, it will enable us to convert the given system of partial differential equations into a new system possessing the same general character, but con- taining one equation less. Upon this the same process may be tried with a similar final alternative — and so on till the required integral is discovered. (On the Differential Equa- tions of Dynamics. Philosophical Transactions, 1863).
CHAPTER XXVIII.
PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.
[THIS Chapter is a reconstruction on a larger scale of part of Chapter xv. At the end of the Chapter reference will be given to other writings of Professor Boole on the subject here discussed.]
1. The general form of a partial differential equation of the second order is
F(x,y, z,p, q, r,s, <) = 0 ............ (1),
where
dz dz d"z d*z d?z
P=^7~> ?=j~» r=j~s> s= J J > ^~T»' * dx * ay ax ax ay a if
It is only in particular cases that the equation admits of integration, and the most important is that in which the differential coefficients of the second order present them- selves only in the first degree ; the equation thus assuming the form
Tt = V ...... ................. (2),
in which E, S, T, and V are functions of x, y, z, p and q.
The most important part of the theory of the solution of this equation is due to Monge, and was extended by Ampere to the more general equation
Rr+ Ss+ Tt+ U(s*-rt)=V ............. (3).
120 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVII I.
This equation, together with the particular equation of Monge, and the equation
both which though falling under Ampere's general form possess peculiarities demanding special notice, I propose to consider in this Chapter. I shall in conclusion make some observations on the theory of partial differential equations of the second order with more than two independent variables.
Monge's method, and Ampere's in so far as it is an exten- sion of Monge's, consists in a certain procedure for discovering either one or two first integrals of the form
~=/(f) ............................. (4),
u and v being determinate functions of x, y, z, p, and q ; arid f being an arbitrary functional symbol. From these first in- tegrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent inte- gration.
Now this procedure involves the assumption that the pro- " posed equation admits of a first integral of the form (4). But such is not always the case. There exist primitive equations involving two arbitrary functions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of the form (2) obtained, but from which it- is impossible to eliminate one function only so as to lead to an intermediate equation of the form (4). Especially this hap- pens if the primitive involve an arbitrary function and its derived function together. Thus the primitive
z = <f> (y + x} + ^ (y - x} - x {f (y + x) - -^ (y -a?)). ..(5), leads to the partial differential equation of the second order
(6),
X
but not through an intermediate equation of the form (4).
It is necessary therefore, not only to consider the case in which the assumed condition is satisfied, but also to notice
ART. 2.]
OF THE SECOND ORDER.
121
what has been done in those cases which do not at present fall under the dominion of any known method.
Genesis of the Equation.
2. PROP. I. A partial differential equation of the first order of the form u =f(v], or its symmetrical equivalent,
F(u,v)=0,
in which u and v are any functions of x, y, z,p, q} always leads to a partial differential equation of the form
Er + Ss + Tt+U(s*- rt] = V.
For, differentiating the proposed first integral with respect to x, and with respect to y, we have
dv
dq
. J>i du du du
-r- \-r + j~P + ~rr + -j-8 du \ax azc dp dq
dFfdv (fa dv_ dv \dx dz" dp
dF (du du du du
du \dy dz" dp dq
dFfolv_ dv
dv \dy dz '
dr 'dp
For brevity, write
'du
da:,
du dx
du >dz
\ /. l*t* Ull/ •, f'-f'-l.\ f €*£
-r- 1 lor - + p -j- , and H- for -j- ^ \J~l J
'du\
JyJ
du fy
flu
and then eliminating
dF dF du ' dv '
|
122 |
PARTIAL DIFFERENTIAL E |
|||
|
we have |
||||
|
(fdu\ |
, du |
L^ol |
\(dv\ |
dv |
|
\(dx)' |
f -j-r- dp |
r ~J~ S( ' dq j |
\\d~y)' |
dp |
|
(fdu\ |
du |
du I |
i (fdv |
dv -r dq
dv dv
r -T -j-nj- -j~r + j- (\dyj dp dq ) (\dxj dp dy
which, on effecting the multiplication, gives
(du fdv\ fdu\ dv] [dp \dy) \dy) dp] r
(fdu\ dv du rdv\ _ fdu\ dv du fdv\ \\dxj dp dp (dx) (dyj dq dq \dy)
(C
du\ dv du fdv^.. dx) dq dq \dx/
du dv du dv \dq dp dp dq
_ fdu\ fdv\ /du\ fdv \dy) \dxj \dxj \dy
a result which, since u and v are by hypothesis given func- tions of a?, y, z, p, q, is seen to be a particular case of the general form (3).
We may hence deduce also the conditions under which particular forms included in the general form (3) arise. Thus, in order that the equation u =f(v] may give rise to a par- tial differential equation of the second order of Monge's form
it is necessary that the condition
du dv du dv _ dqr dp dp dq
should be identically satisfied. This requires, by Chap. IT.
ART. 3.] OF THE SECOND ORDER. 123
Art. 1, that u and v, considered as functions of p and q, should not be independent.
3. The geometrical relations of the equation (3) are also remarkable. It may in particular be shewn that an equation of this form will be satisfied by the equation of any surface which constitutes the envelope of any system of surfaces formed by the variation of three parameters in subjection to two arbitrary conditions. For let the common equation of the enveloped surfaces be
the parameters a, b, c varying in subjection to the conditions
conditions which, determining b and c as functions of a, may be reduced to the form
J = 0(a), c = +(a] ................... (9).
Now the values of p and q being the same for any point in the envelope as for the same point in the generating surface, we have for all such points
_ df(x, y, a. b. c} _ df(x, y, a, b, c)
dx d
These two equations in conjunction with (9) enable us to determine a, b, c as functions of x, y, z, p, q. Let these values be
a = u, b = v, c = w.
Then substituting in (9) we have
equations which hold for all such points. These are then the partial differential equations of the first order of the envelope.
Xow each of these equations is of the general form (4) ; whence by Prop. I. the partial differential equation of the second order is of the form (3), as was to be proved.
124 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
Let us actually construct this equation.
Differentiating the first of the equations (10) with respect to x and to y, and regarding therein a as a function of those variables, and b and c as functions of a, we have
r = d'f i f d*f [ a*f db t d*f dc\da dx" \dadx dbdx da dcdx da) dx '
S=^L+(^L d*Ld± d*f_dc\da dxdy \dadx dbdx da dcdx da) dy '
from which we readily derive
/ d*f\ da ( d*f\ da
I n* _ ^ _ 11 1 _ __ j o _ __ *f _ j _ _ — Q
\ dx2 J dy \ dxdy) dx
Proceeding in the same way with the second equation of the system (10) we have
_
_
dxdy dy \ dy* dx
Hence, eliminating -v- and -j- , we have
/ rfVV f d*f\(f d*_ \S~dxdy) V ~dx*
..-__
dy* dxdy dx* dx* dy* \dxdy) '
the equation sought.
Comparing this with the general form (3) we have the equa- tions
_2 _
dy* _ dxdy _ dx* __ 1 _ dx* dy* \dxdij 1
~r"~u:~ "T"" "
ART. 4.] OF THE SECOND ORDER. 125
d'f d*f tff
whence eliminating ^«, -=i, and —-,- we amve at the 1 dxr dy dxdy
equation,
This then is the condition which must be satisfied in order that the equation (3) may admit of an integral representing the envelope of a system of surfaces in which three parameters vary in subjection to two connecting conditions. It is only proved however to be a necessary, not to be a sufficient, con- dition.
Solution of the equation Rr -f Ss -f Tt + Z7(s* — rt) = V, when a first integral of the form F(u, v) = 0, exists.
4. In the following sections we propose
1st. To shew that when a first integral of the above form exists, its discovery depends upon the solution of two simul- taneous partial differential equations of the first order re- solvable into linear equations.
2ndly. To shew how from such first integral or integrals the second integral is to be obtained.
PROP. II. If the equation
Mr+Ss+Tt+U(s*-rt)=V
admit of a first integral of the form F(u, v) = 0, in which u and v are functions of x, y, z,p, q, then will F(u, v) considered as a function ofx, y, z, p, q, and represented as such for brevity by F satisfy the two partial differential equations of the first order,
3
dq \dy J dp
+ rd/d/
dp dq
126 PAETIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
(fdF\ dF (&F\ dF}
\\--j-] -j- + \-I-\-T-\ =o,
\\dxj dp \dy J dq)
in which
fdF\ dF dF fdF\ dF dF
-j- ) = -r- + P~r> -J~ ) = T" + 2 T- • \cte/ ax L dz \dy J dy * dz ,
Regarding the function F in the proposed integral F= 0 simply as a function of x, y, z, p, q, we have
fdF\ dFr dFg = Q
\dx j dp dq
J- (11).
(dF\ dF dF '
i — j — ) -j — j s ^ — j— t — u \dy j dp dq
On the other hand, regarding F as a function of x, y, z, p, q, mediately through u and v, we have the system
dF (fdu\ du du | d]? ((dv\ dv dv du \\dxj dp dq } dv \\dxj dp dq
dF (fdu\ du du .} dF (fdv\ dv dv '
-T-^r j- ^r(j- ^r ^r du \\dyj dp dq j dv [\dyj dp dq
and these systems are equivalent.
dF Now if from the second of these systems we eliminate -j-
and -7—, we obtain (Art. 2), a result which must be equiva- lent to the proposed partial differential equation,
(13).
This equation then considered as a relation between r, s, t, must be an algebraical consequence of the relations (12), and
ART. 4.] OF THE SECOND ORDER. 127
therefore of the equations (11). If then we determine alge- braically two of the quantities r, s, t, (we select r, t} from the system, and substitute their values in (13), that equation ought to be satisfied independently of the value of the re- maining quantity s. Now supposing p and q to be both con-
. . dF dF
tained in F, so that neither -y- nor ,— vanish, we have
dp dq
from (11),
dF\ dF
fdf\ dt_
dF dp
tdF\ dF \dy J dp ""
dq
substituting which in (13) there results
fdF\ dF (dF\ dF (dF\ (dF\ dF dF K L* J ^V~
{?? (dJL } \dq
_ R ^ — T ( d— Y dq dp \dp /
Now as this equation is to be satisfied in virtue of the con- stitution of By S, T, U, V, and the function F, and indepen- dently of s, both the coefficient of s and the absolute term not containing s must be separately equated to 0. Thus F con- sidered as a function of x, y, z, p, q, and containing p, q, at least must satisfy the partial differential equations
fdF\dF (dF\dF
•" ~r~ ~i — \r J- [-y- \ ~r
\dx J dq \dy J dp
dp
dF dF
dF\dF fd_F\dF\ +~
(15).
128 PARTIAL DIFFERENTIAL EQUATIONS [OH. XXVIII.
This result may also be established by forming the equa- tions of condition which express the proportionality of It, S, ...V, to the corresponding quantities in the constructed equation (7). From these equations of condition it is actually possible to eliminate in two distinct ways the quantities
(dv\ (dv\ dv dv , , , . , f . c
I -3- i [~r-]t -7-5 -j- •> the result being the formation ot two \dx) \dyj dp dq
partial differential equations for u agreeing in form with those above given for F. (See the memoir Ueber die partielle Diffe- rentialgleichung . , . Crelle's Journal, Vol. 61.) The actual transition from the former to the latter rests upon the con- sideration that the equation F (u, v) = 0, when F is arbitrary, is not really less general than the form <1> {F (u, v), v} = 0, in which the <5> is arbitrary. And here u has been replaced by F(u,v).
The only condition respecting the application of the above equations is that we do not admit any relations which make
... dF dF .
either -7— or -7- to vanish. dp dq
5. PROP. III. The solution of the system of partial diffe- rential equations established in the last proposition may in all cases be made to depend upon that of simultaneous linear partial differential equations of the first order.
In demonstrating this proposition we shall consider first the case in which JJ— 0, then the case in which V= 0, lastly the case in which neither of these quantities vanishes. The ground of this division will appear in the investigation.
Case 1. Suppose U= 0. The equation then is of Monge's form,
Hr+Ss+Tt= V.
The second equation of the system (15) becomes
ART. 5.] OF THE SECOND ORDER. 129
and therefore breaks up into the equations dF dF dF dF
~1 mi -J- = °> -J m-2 J ~ = °>
dy 1dp dq 2dp
m1 and m2 being the roots of the quadratic equation
Bm*-Sm+T=0 (16).
As each of the above constituent equations is of the form
dF dF -j- =m ~j— , dq dp
the system (15) may be reduced to the form
Em (—} — T (—} ~ -u Ym — — = 0 \dx J dp \dy J dp dp dp
which breaks up into the equations
j — vj •*-* *** I _7 I ' l 7 1 1^ " * 7 — *
dp \ax J \at/ J dp
The former of these we must reject (Art. 4). There re- mains for the determination of F the system of linear partial differential equations
**^m££m'i]
da dp
j- (17),
- Vm -j- = 0 I dp J
and there will exist either one or two systems included under this form, according as the roots of the quadratic (16) are equal or unequal.
B. D. E. ii. 9
130 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
Case n. Let F = 0. The system (15) then becomes
fdF\ dF (dF\ dF fdF\ (dF\ _ '} L ^ U -
dp TT((dF\dF fdF\ dF} .
+ u\ w-hr- + ( T- -j-t = Q'
[\dxjdp \dy J dq )
Eliminate Vby multiplying the first equation by
fdF\ dF^ fdF\ dF \dx) dp \dy J dq '
the second by
fdj\ (dF\ \dx)\dy)>
and subtracting; we obtain, after rejection of the common
, dF dF factor -j- -j- , dp dq
dF
We shall put this equation in the place of the second equa- tion of the system. This we are permitted to do under the restriction that in seeking to satisfy the system so changed we do not make use of any relations which would cause either of the two factors employed in the process of elimination to vanish or become infinite.
The new equation reduces to one equation, or breaks up into two equations of the form
©-©-
m being determined by the quadratic equation
Sm+ T=V.
A1IT. 5.] OF THE SECOND OEDEE. 131
/ '/•] 77\ //7 7it\
in (-— = m--
Making -^— = m-j-j ^ tne fi*8* equation of the system (15), we get
dF\( dFdF dF\
which breaks up into
But if we combine the first of these with (18), we obtain
(D=°> ©-*
and this combination causing both the factors employed in the elimination of U to vanish must be rejected. There remains then the combination
(19),
and this will represent either one or two systems of equations according as the quadratic determining m has equal or un- equal roots.
Case m. Let neither £7=0 nor F=0.
Multiply the second equation of the system (15) by an indeterminate quantity I, and add to the first ; then we have
J dp \dy J dp mfdF\dF fdF\dF
Ul [ -j- I -j — \- M [-J- } -j- \dyJ dq \dxj d%
9—2
132 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXY1II.
We shall enquire whether it is possible so to determine I as to resolve this into linear factors.
We might investigate this by resolving the equation as a quadratic with respect to -j- or -y-. But the form of the equation suggests what the forms of the linear factors must be if the resolution be possible. For as the squares of -3—
and -j- both appear, and these squares alone, in the func- tion to be resolved, it is clear that -=- and •T- will be the
dq dp
only differential coefficients of F which will appear in both linear factors in common. The most general supposition
TTfl JTjl
possible is then that one factor shall contain -7- and ^- with
dq dp
- | , the other the same with ( -3- ) .
far \»//
Assuming then one factor to be of the form
,dF dF fdF\
I -j- + m -j- + n \-j~ I ,
dq dp \dxj
it is seen from the. form of the coefficients of the first three terms of (20) that the other factor must be of the form
„ dF Tl dF U fdF\
J£ I L / J
dq in dp n \dy J ' and the resolved form of (20) must be
U^? W^.V/<ffi\UldF+mW+n/d^ =()
{ dq m dp n\dy}}\ dq dp \dx))
Multiplying out and equating coefficients, we obtain the conditions
ART. 5.]
OF THE SECOND ORDEE. Tin
n
ui-ul IT'
133
TP
m
The third and fourth of these conditions are equivalent, and give n = 1. The first and second are also equivalent,
T and give m = y%. These values reduce the last equation of
condition to
so that I is determined by a quadratic. The resolved form of equation (20) now becomes
1 7? dF± TTl dF^ TT (dF\\ \1 dF^ T dF^ (dF\\ Iti j- + ill j — F- U f-y- }[ \l -j- + -Tf -}-+ (-j- lr = $•
[ «2 "p \dy/) { d% U dp \d
To these results Tve may give a somewhat simpler form by making Ul=m; not the m used above. We have then as the quadratic for determining m,
- £77=0
(21),
and as the resolved form of (20),
&* m*£±nl**\\l dF.i.TdF+TT(d*'}\-0 -J-T + in -j— +1/1-5- r \ m T~ + •• j r ^ V T7~ / r "•
^2. ^P >^ ^J 1 d<l dP
134 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
Let ml and m2 be the values of m. Then we have from the last the two distinct equations
\B**+& + u(d/}\L'lf'+T^ + U(d/}\ = 0,
{ aq 1 dp \dy J } [ 1aq dp \dxj)
dF Tj(dF\\ ( dF_ Td,F n (dF\\_ dp \dy J ) \ 2 dg dp \dx )}
and it is evident that these will be together equivalent to the equations (15) from which they were derived.
Now to satisfy these equations simultaneously it is neces- sary that we should equate to 0 one linear factor from each of their first members. If we equate to 0 the first linear factors, we have
dF dF fdF\ d([ 1 dp \dy )
dq 2 dp \dy J
whence, by subtraction,
.dF
This combination must therefore be rejected (Art. 4). For the same reason must the combination formed by equating to 0 the second linear factors in the left-hand members of the above two equations be rejected. There remains then only the combinations formed by equating to 0 the first factor of one of these members, and the second of the other.
Thus we should have the combination TdF dF TT(dF\
<fy m*3p ^ (22^
dF ^dF ^(di
ART. 7.] OF THE SECOND ORDER. 135
with the combination which would be obtained from this by interchanging m^ and mt.
6. It results from the foregoing investigations that the function F is in all cases to be determined by the solution of two simultaneous linear partial differential equations with five independent variables. Xow the theory developed in Chapter xxv. shews that the number of integrals of such a system cannot exceed three. That theory enables us both to determine what the number of integrals is. and to construct the system of ordinary differential equations, reducible to the exact form, upon which their discovery depends.
We have seen that the knowledge of two integrals u = a, v = b of the system enables us to construct a general first integral
of the partial differential equation (3). And the solution of this first integral would lead us to the second integral which is the final object sought. But the direct solving of a partial differential equation of the first order which is not linear and which involves in its actual expression an arbitrary function is difficult, and happily it may be avoided here. The fol- lowing propositions will enable us to accomplish the virtual solution by a different solution, founded however upon the same general principles.
7. PROP. iv. The integrals of the respective systems of simultaneous linear partial differential equations upon which the determination of F depends are so related that if from tiro such respective integrals the values of p and g_ are determined, they will render the equation
dz =pdx + qdy
integrable. And in the particular case in which the tico systems become identical, any two integrals of the system stand in the same relation.
136 PARTI AL- DIFFERENTIAL EQUATIONS [CH. XXVIII.
For, let <I> be an integral of the system (22), and "SP an integral of the associated system obtained by interchanging ml and 7W2 in the case in which these quantities are different. Then <I> satisfies the equations
i
2 ay dp
and W satisfies the equations
' dV , dV TTfdV\ R -j- + mz -r- + U ,- = 0, dq * dp \dy J
^-j- -j- 1 dq^ dp
But the necessary and sufficient condition in order that the values, of p and q derived from the equations <E> = 0, M/1 = 0, may render dz —pdx — qdy integrable, is
dx J dp dp \dx .
r Jfa //7\I/\
= 0 (23).
dy. See Chap. xiv. Art. 11, Equation (36).
Now if from the previous equations we determine the values of
V7 J J 1 7 I j 1 ~~? I * I -T" I « dx) \dy J \dxj \dy J
and substitute them in the above equation of condition it will be identically satisfied.
ART. 7.] OF THE SECOND ORDER. 137
The determination of [-^-] .... from the previous systems \dxj
requires that U should not vanish. Hence the proposition is established except in the case of £7=0, which is left doubtful.
To examine this case let us revert to the system (17) which is proper to it. To that system since
whence Smjn^ = Ty
we may give the form
dF_ dF_ fdF\ fdF\ V dF_
dq l dp \dx J * \dy J R dp
or the form obtained from this by interchanging m1 and ma.
Substituting in these respective forms <I> and ^ in succes- sion for F, we find
d$> d&
_ V dV \dx ~ > \dy B dp '
and these values substituted in (23) reduce it to an identity. Thus the proposition is established generally.
Lastly, as in the case in which the two roots of the quad- ratic for determining m are equal, the two systems of partial differential equations for determining 4> and ^ become one, it follows that if from two integrals of that one system we can deduce values of p and y these values will render the equation,
dz — pdx — qdy = 0 integrable.
138 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXYI1I.
8. PKOP. V. When the system of simultaneous linear par- tial differential equations determining F admits of two integrals u = a, v = b, it will admit or will not admit of a third inte- gral w = c, according as the roots of the quadratic determininy m are equal or unequal.
The system in question, (22), becomes when we divide by
rr J -x t fdF\ , fdF\ A, . , „
U and write for — =— and -,— their full expressions \dy ) \dxj
dF dF m^ dF It dF _ dy+% dz+ U dp* U~dq~
dF dF TdF ™*dF_ dx+P dz + U dp+ U dq~ '
or At^=0, A2F=0,
in which
d m d R d
d d T d mz d + Udp+ U dq
Hence the equation
(^ becomes
m —m dF f T
//€-, *'*o *«* / * JL
In this expression the coefficient of the first term only has been calculated.
Now, by the theory developed in Chap. xxv. in order that the two simultaneous partial differential equations should have their full complement of integrals (three) it is necessary that the above equation should be satisfied identically. This involves three conditions, namely,
\RT. 8.1 OF THE SECOND ORDER. 129
the first of which is the one affirmed in the Proposition to be
necessary.
Secondly, it is to be shewn that if this condition be satis- fied and it the system of given linear equations admit of two integrals u — a, v = b, it will admit of a third.
Keplacing ml and m3 by m the system becomes dF dF m dF £ dF_ dy + 2 dz + U dp + U dq ~
dF dF T dF mdF_ dx+Pfc + U dp + U dq =
Now if we construct from this the corresponding system of ordinary differential equations, we shall find it to be
dz —pdx — qdy = 0, dp- ^dx-^dy^O,
dq - ^jdx - jjdy = 0.
Xow it is impossible that the first of these equations should be integrated without a previous determination of p and q as functions of x, y, z, seeing that dx, dy, dz are the three differ- entials entering into that equation. Such determination can only come from the integration of the second and third equa- tions of the system. But if these equations can be integrated in the forms u = a, v = b, then u and v being particular values of F satisfying the partial differential equations, it follows from the last Proposition that the values of p and q which they will yield will make the first equation integrable. Hence if the system admits of two integrals it will admit of three ; as was to be shewn. On the basis of these Proposi- tions the theory of the second integration rests.
140 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
Theory of the Second Integration. 9. First suppose the values of m unequal.
Then MI = al, vl = bl "being the two integrals (and we have seen that there cannot be more than two) of one of the systems of linear partial differential equations, and u2 = aa, vz = bz those of the other, the general first integrals of the given system will be
The values of p and q determined from these will by Proposition iv. render
dz —pdx — c[dy = 0
integrable, and the integral of this will be the general integral of the proposed partial differential equation. For it will in- volve explicitly or implicitly two arbitrary functions derived from those in the first integrals.
It suffices however, following herein Charpit's method, to combine one general first integral derived from the one sys- tem with a particular first integral derived from the other system, e.g. the integrals
The values of p and q hence derived, and employed as before, will lead to a second integral involving one arbitrary function and containing two arbitrary constants. This con- stitutes a complete primitive from which the general solution will be obtained by converting one of the arbitrary constants into an arbitrary function of the other, and eliminating the latter between the equation and the one derived from it by differentiation with respect to that constant.
Secondly, suppose the values of m equal.
In this case we have but one system of partial differential equations so constituted however that if it admits of two inte- grals it will admit of three.
ART. 9.] OF THE SECOND ORDER. 141
Let u = a, v = b, w=c represent these integrals. Then if from these we eliminate p and q we shall obtain a final inte- gral of the form
.and this constitutes a complete primitive from which we shall deduce the general integral by making b = <f> (a), c =i|r (a), and eliminating a between the equations
at \x. y. a.
f\ J \ * •* '
0 =
da
To prove this let us combine the general and particular first integrals
v = ^> (M), u — a.
The values of p and q hence obtained make
dz — pdx — qdy = 0
integrable, and the result can be no other than the remaining integral w = c, or rather what this would become on eliminat- ing^? and q from it. But since the equations by which this integration are to be effected are equivalent to
u = a, v = <f> (a),
w will become a function of x, y, z, a and <f> (a). Also by Charpit's method c is to be treated as a function of a, so that ultimately we have the result above assigned.
have here supposed U not to vanish. If it do the theory assumes another but simpler form. Let
be the two general first integrals. Then, since by the con- dition at the close of Art. 2, if p be eliminated from these equations q will also disappear, it suffices to eliminate them together in order to obtain the general second integral.
142 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.
10. Although the cases in which U= 0 and V= 0 have in the foregoing sections been treated for simplicity apart, their theory might have been deduced from that of the case in which neither ?7rior V vanishes.
Thus to deduce the equations for the case of U= 0 elimi-
3TJ1 TTfl
nate from the general system (22) -y- and -j- in succes- sion, and we find
.dF TT fdF\ rr* (BT_ m^ _ _ ^ (_) +
(RT - »X) ~- + UT( ~ j - Um^ ( ~ } = 0.
But from (21) RT- m,mt = UV. Substituting, and then dividing by ?7we find ~dF
>dF^ the equation determining ml5 mz being
This is equivalent to the results of Art. 5, Case I.
11. We found it necessary (Art. 3) in order that the gene- ral partial differential equation of this Chapter should be satis- fied by the envelope of a system of surfaces the equations of which contain three parameters varying under two conditions that the relation
should be satisfied.
It appears from Art. 8 that this is but one of three condi- tions necessary and together sufficient for this purpose. The formal conditions for every form of ultimate solution con- sistent with the existence of a general first integral F (u, v) = 0 can be deduced in the same way.
CH. XXVIII.] OF THE SECOND ORDER. 143
[In the Bulletin de VAcademie Imperiale des Sciences de St P/lersbourg, Vol. IV. 1862, there is an article entitled Con- siderations sur la recherche des integrates premieres des equa- tions differentielles partielles du second ordre, par G. Boldt (Lu le 7 Juin 1861).
The article occupies pages 198 — 215 of the volume. Al- though the name does not quite correspond, I consider that to be a misprint, and I attribute the article to Professor Boole, partly from the nature of the contents, and partly because it is known by his friends that he was engaged at a time corre- sponding to the date here given in the preparation of a mathe- matical article in French.
The object of the article is to determine the conditions necessary for the existence of a first integral of the equation
^ d*z ^ d*z ™ d'z
where R, 8, T, and TFare any functions of x, y, z, ~ and — ;
a x ay
and also to determine the conditions which must hold in order that Ampere's method of integration may be employed.
In Crelle's Journal, Vol. LXI. there is an article by Pro- fessor Boole, entitled Ueber die partielle Differentialgleichung zweiter Ordnung Rr + Ss + Tt + U(s* - rf) = V.
The article is dated 1862 ; it occupies pages 309 — 333 of the volume.
Among Professor Boole's manuscripts I found a memoir very closely resembling the article in Crelle's Journal; it
144 PARTIAL DIFFERENTIAL EQUATIONS &C. [dl. XXVIII.
would appear that the memoir was drawn up with a view to publication in the Transactions of some English Scientific Society, and that this design was afterwards abandoned in favour of the article in Crelle's Journal.
After some hesitation I have resolved to print this memoir. Even if the memoir had been identical with the article in Crelle's Journal it would have been convenient to the English reader to be able to avail himself of the investigations ; and the memoir contains remarks which do not occur in the article, and which are interesting in connexion with the history of the subject. There is some repetition of matter which has already been given in Chapter xxvui.; but I was unwilling to impair the completeness of the memoir by abridgment or omission. Accordingly the memoir forms the next Chapter of the present volume.
In Article 2 of the next Chapter will be fo