UC-NRLF
*B 532 flb3
THE DOCTRINE OF GERMS,
OR
THE INTEGRATION OF CERTAIN PARTIAL
DIFFERENTIAL EQUATIONS WHICH OCCUR IN
MATHEMATICAL PHYSICS.
BY
S. EAENSHAW, M.A.,
AUTHOR OF "ETHERSPHEUES A VERA CAUSA OF NATURAL PHILOSOPHY.
DIVERSITY)
CAMBRIDGE:
DEIGHTON, BELL, AND CO.
LONDON: GEORGE BELL AND SONS
l88l
CamfrrtUge :
PRINTED BY 0. J. CLAY, M.A.
AT THE UNIVERSITY PRESS.
PEEFACE.
The method of integration by means of Germs adopted in
this Treatise is based on the admitted principle that in the
work of integrating a proposed differential equation we are
free to avail ourselves of the advantages offered by any dis-
tinctive peculiarities that are perceived to exist in the equation
itself prior to its integration. Any such peculiarity will, as
a matter of course, impress a corresponding peculiarity on
the integral to be found. Equations will therefore be classified
according to their distinctive peculiarities, those peculiarities
being indicated by the particular ways in which germs may
be connected with the variables of a differential equation
without disturbing or in any way affecting its form.
In this Treatise the differential equations that will be
brought before the reader are all linear and partial, in con-
sequence of which the doctrine of germs suitable for such
equations admits of being presented in a form that is easily
reduced to a system of singular efficiency. But there are
certain other equations that are not linear, and which there-
fore do not fall under the system that will be developed in the
following pages. The equation t-^J -J^ = -t^ is one of this
kind, for its form will not be affected if ax be written for x ;
neither will it be affected if bu and bt be written simul-
taneously for u and t ; and these arbitrary constants a, 6, will
therefore necessarily find a place (either explicitly or implicitly)
in its integral. Also as only x takes the constant a, this
IV PREFACE.
indicates a peculiarity of x as to the manner in which it can
appear in the integral. That u and t are alike related to b
is indicative of the existence of a peculiar relation of u to t.
And, further, we may write u 4- A for u, x + g for x, and t + h
for t (the constants A, g, h being perfectly arbitrary) without
affecting the form of the equation. This shews that if U be
an integral of it, so likewise will be the following,
gd hd
edx+dt (JJ+A).
We may therefore presume that as there has been found for
equations that are linear a " Doctrine of Germs" so there may
be a possible "Doctrine of Germ-like Constants" for equations
that are not linear.
In Chap. III. is introduced a theory of "Symbolical
Equivalences." The subject is regarded from a point of view
which may be considered as in some degree new. The exigencies
of this Essay did not seem likely to require the complete
development of this Theory ; and in consequence of this only so
much detail is given as was likely to be wanted in subsequent
chapters. The Theory is capable of throwing light on several
troublesome known paradoxes which have often been a source
of perplexity to the Mathematical Student.
As the author was induced to undertake the development of
the Doctrine of Germs by a desire to accomplish the complete
integration of Laplace's Equation, and the consequent discovery
of the general form of Laplace's Functions, he has deemed it to
be of some possible advantage to obtain and to exhibit those
results in several forms and aspects ; hoping also that by this
diversity the reader would be the more disposed to accept
results which so confirm one another.
It has been taken as an admitted definition of Laplace's
Functions, that any quantity which satisfies the equation (1) of
PREFACE. V
Art. 127, is a Laplace's Function. Laplace's Equation is usually
given in two forms, viz. those in Arts. 124, 127 ; and to both of
these results have been adapted. The first of these forms does
not contain r\ the second does; and it will be seen that r
enters the latter in a manner that demands peculiar manage-
ment, and when so managed leads to remarkable results, which
can hardly fail to throw some light on the usually received
theory of Laplace's Functions.
There appears to be an exceptional case of these functions
which the reader will find in Art. 131.
It is needful to advise the reader before he enters upon the
task of reading some parts of this Treatise, that when arbitrary
constants occur in a general integral of a linear equation it will
consist of the sum of several subgeneral integrals ; and each of
these constants being arbitrary by nature will retain their arbi-
trary character when multiplied by a definite numerical quan-
tity ; and if such a constant be separated into several arbitrary
parts, each part will be as arbitrary as the original constant.
Hence in altering the form of a general integral it will not be
necessary to preserve the identity of any such constants, but
only the quality of their independent arbitrariness; and thus
we need not observe with respect to them the usual require-
ments of algebraic rules of reduction in passing from step to
step. The arbitrary constants referred to are used merely to
indicate the absolute independence of the subintegrals or sub-
general integrals of which they are the respective coefficients.
(See Art. 42 for an example of what is here alluded to.)
The author has pleasure in acknowledging the valuable
assistance rendered him by Mr Greenhill, M.A., Fellow of
Emmanuel College, in supervising this Essay in its passage
through the press.
Sheffield, Jan. 1, 1881.
Digitized by the Internet Archive
in 2008 with funding from
Microsoft Corporation
http://www.archive.org/details/doctrineofgermsoOOearnrich
TABLE OF CONTENTS.
CHAPTER I.
PAGE
Introductory Remarks 1
CHAPTER II.
General Properties of Germs 8
CHAPTER III.
Symbolical Equivalence 28
CHAPTER IV.
Transformation of Linear Differential Equations ... 36
CHAPTER V.
Integration of Reduced Forms ; Two Independent Variables . 41
CHAPTER VI.
Equations nearly related to Laplace's Equation .... 53
CHAPTER VII.
Integration of Equations of Three Independent Variables . . 73
CORRIGENDA.
Page 22, line 6; for subsequent read subgeneral.
,, 25, line 3; for +c'z) read +c'z)u.
,, 29, line 8 ; for algbraio read algebraic.
,, 46, line 11 ; for eMf> read eMH.
THE INTEGRATION
OF
LINEAR PARTIAL DIFFERENTIAL EQUATIONS.
V OF THE
(ufjte-
'\tf>:
CHAPTER I.
INTRODUCTORY REMARKS.
In the present work we have not to deal with Linear Partial
Differential Equations in general, but only with such as are
known to be of difficult integration, and which have been found
to present themselves in connexion with the application of
Mathematics to various branches of Natural Philosophy.
This task we have undertaken not as a branch of analytical
enterprise, but as a contribution to the resources of those philo-
sophers who think it a matter of importance that Physical
Theories should be subject to the severe test of mathematical
confirmation. And in the execution of this task we believe that
we shall have the privilege of developing a method of integra-
tion which may be regarded as new, and that is singularly well
adapted to the integration of certain equations which have been
found intractable by ordinary methods.
1. In some cases a remarkable degree of uncertainty and
intricacy besets the answer to the question, — when may an inte-
gral of a linear differential equation be rightly styled its general
integral ? The following are some of the reasons of this uncer-
tainty.
E. 1
2 INTRODUCTORY REMARKS.
If one of the independent variables (as x) of a linear differ-
ential equation occur therein not as a symbol of quantity but
only as a symbol of differential operation (as -y-J , then (suppos-
ing U an integral of the equation) not only will U be an integral
but so likewise will each one of the quantities
AdU nfU rd*U ....
Adi' Bdx*' ^,.... ad infinitum,
and (speaking generally) we thus can out of a single known
integral create an unlimited number of new integrals all differ-
ent from the original and from each other.
And not only can we thus create new integrals, but out of
these new ones we can by addition of all of them, or of a few of
them selected arbitrarily, create an unlimited number of fresh
integrals different from U and from each of those previously
found from U by differentiation.
Now as we have the right of forming known integrals into
groups as we please, and from the nature of the integrals of a
linear equation each group will be a new integral, we are obliged
to come to the conclusion that it is not easy to see on what
principle we can say of some one of the infinite mass of integrals
a proposed linear differential may have, that it is the general
integral.
2. We shall be able to shew, in the case of every linear
differential equation of the class supposed in the preceding
article, that it admits of integrals of a peculiar kind, forming a
distinct class, and they are generally infinite in number, and
when added together form a sum which is equivalent to the
general integral.
For distinctness of reference we shall denominate these in-
tegrals subiutegrals ; and when we speak of those collectively
which belong to the same differential equation we shall denomi-
nate them a family of subiutegrals.
If then P, Q, B, S,... be the individual members of a family
of subiutegrals, and u be the general integral of the same equa-
INTRODUCTORY REMARKS. 3
tion as that to which they all individually belong, we shall
represent the relation between u and P, Q, P,... by the following
equation,
u = AP + BQ + CB + DS+... adinfin (1),
A, B, C, D, ... being arbitrary constants, the use of which in
this equation is, to indicate the absolute independence of
P, Qy R, ... as integrals of the proposed equation.
3. Sometimes the general integral (1) will divide itself by
some peculiarity of form into two or more distinct parts; and
to these independent parts we intend to refer under the desig-
nation of subgenera! integrals.
The number of such subgeneral integrals that belong to a
proposed differential equation is generally dependent on the
order or some other peculiarity of the equation.
4. When we know a family of subintegrals (as P, Q, P, ...)
we can by grouping them into different heaps, and finding the
sum of each group, take the various sums thus formed as a
family of subintegrals; and as a family it will be symbolically
equivalent to the original family (P, Q, P, ...), though the
members of the new family may happen to have no similitude
of form to the members of the other.
Their equivalence results from the one fact that each of
them explicitly or implicitly contains all the members of the
subintegral family of which u (the general integral) is known to
be constituted.
Thus a family of subintegrals always admits of being recast
by grouping, by summation of series, and other means whereby
a change of the forms of its members is effected.
This is important because it brings forward the question, —
what will be the most convenient forms in which a family of
subintegrals can be obtained ?
One answer to this would be; — let the members of the
family be cast in such forms, that if any one member of the
family can be found all the other members may be obtained
1—2
4 INTRODUCTORY REMARKS.
from it by simple and repeated differentiations or integrations
of that one member with respect to one or more of the inde-
pendent variables contained in the proposed differential equation.
We shall in due time shew how and under what conditions
this may be done.
5. As the existence of subintegrals is but little known, we
shall here add the following illustration.
d2u du
Let -7-2 ==-7- be a differential equation; then we can at a
glance detect the following as independent integrals of it :
' V i.2 + y' l.a.sTi:!1 1. 2. 3. 4"1"!.. 2.1 1.2'
each of which can be obtained from the one before it by inte-
gration with regard to x, and correction with regard to y. If this
be the whole family then
A, B, C, ... being arbitrary and absolutely independent con-
stants.
G. We shall see in future articles that changes of the inde-
pendent variables of a proposed differential equation can be
sometimes made without producing any effect whatever on the
form of the equation.
Whenever this can be done, the same changes of the same
variables may be made in any known integral of the proposed
equation without depriving that integral (however much changed
in form thereby) of its property of being still an integral, or of
its generality as an integral.
Also if the known integral should happen to be a particular
and not a general integral the change of variables just described
would introduce such a change of form of the integral itself as
might bring it nearer to the form of a general integral by intro-
ducing new arbitrary constants which we should be at liberty to
treat as germs not existing in the original particular integral.
INTRODUCTORY REMARKS. 5
7. We have just used the word germ; let us now explain
what we mean by it.
We are aware that integration generally introduces to our
notice in the integral certain constant quantities which have no
existence in the differential equation itself. Such constants are
in fact the offspring of integration ; and are generally denomi-
nated arbitrary constants. The use of such constants in prob-
lems is well known.
This designation however is not sufficient for our purpose,
and we intend to speak of them, under certain circumstances, as
germs, or germ-constants. For as each of the variables of a pro-
posed linear differential equation is constant with reference to
every operating differential symbol contained therein except
its own, so an arbitrary constant (germ) is constant with refer-
ence to all the differential symbols except its own; and it or
any function of it contained in u may be operated on by its
own symbol of differential operation, though no such symbol is
contained in the proposed equation.
We may therefore consider a germ as being a new independ-
ent variable, i.e., an independent variable that is not contained
in the differential equation itself, but only in its integral.
Thus €ax+a7y is an integral of the equation -^ = -=- ; but the
constant a is constant only in reference to -7- and -=- , but not
ax ay
3
in reference to -y- ; and therefore in this integral we may con-
sider a either an arbitrary constant, or a new independent
variable additional to x and y; and this is the property to
which we refer when we call a a germ.
8. We shall find it convenient to be able to speak of certain
germs under specific names, which will refer to the manner in
which a germ in an integral may happen to stand (actually or
virtually) connected with its independent variable. Thus if a
germ g and an independent variable x stand connected in an
integral by addition (as distinguished from multiplication), (as
in the form x±g) we shall refer to g as the minor germ of x.
6 INTRODUCTORY REMARKS.
But if the connexion be of the nature of multiplication
(as gx or -) we shall speak of g as the major germ of x.
9. Germs may also be regarded as being general, or real.
A general germ may receive any value whether real or
imaginary.
A real germ may receive only such values as are not imagi-
nary.
Nevertheless a general germ may be perfectly represented by
means of two real germs. Thus if K be a general germ, the
equation
K=M+im,
in which M and m are independent real germs, will perfectly
represent K, Hence a general germ is equivalent symbolically
to two real germs.
We shall throughout this Treatise use the two quantities i
and j in the ambiguous senses implied by the two independent
equations following,
i* = — 1, and j* = + l;
and both i and j will be regarded as independently carrying
with them their proper double algebraic signs.
10. There are functions of x which cannot be expanded by
MacLaurin's Theorem; and therefore the series A +Bx+ Cx*+ . . . ,
in which the coefficients A, B, C, ... are all arbitrary, does not
symbolically represent a perfectly arbitrary function of x; but
Axa + BaP-\- Cxy+... in which A, B, G, ... are arbitrary and
a, j3, 7, ... not limited by the condition that they are to be posi-
tive integers, symbolically represents a perfectly arbitrary func-
tion.
We may distinguish these cases, when necessary, by denomi-
nating the former a MacLaurin's arbitrary function.
11. To find the potentiality of a germ when it occurs in an
integral only as an index or power of a function of the inde-
pendent variables.
INTRODUCTORY REMARKS. 7
Let Wcom be an integral of a proposed equation, W and co
being functions of the independent variables, and m being a
germ that occurs only as the index of the quantity denoted by co.
We may give to m an infinite series of different values
a, /3, <y, ... at pleasure; and each one of these values of m will
furnish us with an independent integral; and all the integrals
so obtained we may unite in a single integral in the following
manner,
Wcom = W (Acoa + Bcop + Ccoy + . . . ad infin.),
A, B, C, ... being independent arbitrary constants.
But a, £, 7, ... being arbitrary also, the series
Acoa + Bcop + Ccoy + ...
will represent an arbitrary function of co of the most perfectly
arbitrary kind ;
.-. Wcom=WF(co).
Hence a germ when it occurs in an integral as an index only
is potentially equivalent to an arbitrary function of the most
general kind.
The converse is manifestly true, viz., if F (co) be a perfectly
general arbitrary function of co, then will F (co) = com symboli-
cally.
But if F(co) represent a MacLaurin's series only, then it does
not follow that F(co) = com symbolically, for in this case F (co) is
clogged with the condition that the powers of co in the expan-
sion of F (co) must be positive integers. For such a function we
shall therefore when it occurs have to find a potential equivalent
clogged with the same condition. com is, as we have said, too
general, and may consequently (if incautiously used) lead us
into error when we come to the generalizing of results obtained.
12. It will be a convenience to be allowed sometimes to
represent the product 1 . 2 . 3 ... n by the symbol n!.
CHAPTER II.
SOME GENERAL PROPERTIES OF GERMS.
13. Let ot.w=0 represent a general linear partial dif-
ferential equation of any number of independent variables, u
being the dependent variable, and zr denoting the compound
operating symbol. Also let U denote any integral of this
equation containing a germ. Denote the germ by c, and expand
£7 in a series according to the powers of c ;
.-. U=Pcp+Qcq + Rcr + (1),
in which p, q, r, ... are definite indices, and P, Q, P, ... are
functions of the independent variables.
Operate on each member of this equation with w, noting
that<*(C0 = 0;
. • .' 0 = («r . P) (f + (« . Q) cq + (*x . R) cr -f . . . .
Now c being a germ is an arbitrary independent variable,
and consequently this must be an identical equation ;
.'. 0 = S7.P, 0=<GT.<3, 0=<G7.P,...
that is, P, Q, R,... are independent integrals of the proposed
equation ; they are, in fact, the family of subintegrals, the
members of which are rendered independent by the fact that
the powers of c in equation (1) are all (Jifferent. They owe
their independence to the presence of a germ in U.
But we can preserve their independence another way, and
at the same time unite the subintegrals in a single integral,
thus
U=AP+BQ + CR + (2),
in which A, B, C, ... are arbitrary constants.
SOME GENERAL PROPERTIES OF GERMS. 9
Comparing this with (1) we perceive that the different
powers of a germ in an expanded integral are symbolically
equivalent to independent arbitrary constants.
We have now obtained the power of eliminating a germ by
expansion of an integral according to the powers of that germ.
And, conversely, we can eliminate arbitrary constants,
which belong to a series each term of which is a subintegral,
by means of an extemporized germ.
14. If the integral U in the preceding article should
happen to contain a second independent germ (n), then as
only m has been eliminated the subintegrals P, Q, R, ... will
each contain the germ n constituting each of them a germ-
integral of the proposed equation. From each of these n may
therefore be independently eliminated, and each of them will
be thereby resolved into its own constituent subintegrals.
Thus we have before us the fact that a subintegral may be also
a germ-integral, and in this character resolvable into subinte-
grals of a more elementary class : and the ultimate subintegrals
are those which do not contain a germ, and into which a germ
cannot be introduced.
We may arrive by one step at the ultimate subintegrals
of U by expanding U in the first instance in a series according
to the powers of both the germs m and n, and then writing
independent arbitrary constants for the germs and their powers
and different combinations.
The converse is also true, viz. that we may eliminate arbi-
trary constants by means of the powers and the combinations
of the powers of two or more independent germs.
Thus if
we may (if A, B, G, ... be independent arbitrary constants)
express a symbolical equivalent to this series by means of two
independent germs m, n, thus,
U=A jl + {mx + ny) + (m* ^ + mn ^ +n2 ^Q + ... j
= A€mx+nv,
10 SOME GENERAL PROPERTIES OF GERMS.
We may reverse this process and at one step assume, if m, n
be independent germs, the following symbolical equivalence,
15. As a matter of experience we know that different
physical and geometrical problems lead us to the same forms
of partial differential equations. The equation
d*u d2u (Pu _ fl
is a well-known example of this. Hence each member of a
family of subintegrals being a complete integral in itself of its
-kind, expresses the solution of a particular problem in physics
or geometry, which can exist independently ; and it also
expresses what may be called an independent elementary state
of matter or some affection thereof; or some imaginable inde-
pendent geometrical condition of an elementary nature. The
subintegrals being independent, represent properties which can
exist independently in nature.
Hence whenever an integral can be resolved into elementary
subintegrals we have in such cases this fact before us ; that the
problem which brought us to the corresponding differential
equation is really of a compound nature and capable of being
resolved into a number of elementary problems, the super-
position of which in their proper proportions is equivalent to
the original problem.
Hence, also, one problem may require for its complete
representation one set (or group) of members selected from the
whole family of subintegrals; and another problem another
set of members. And thus comes into use that important
property of all linear differential equations, that the sum of
any of the members of a subintegral family is an integral of
the same differential equation and represents the superimposed
action of so many different geometrical or physical properties.
The selection of the group of individual members of a
family of subintegrals suitable for the solution of a given
SOME GENERAL PROPERTIES OF GERMS. 11
problem is sometimes a work of difficulty, and will always
make a demand upon the investigator's ingenuity.
1G. The connexion between the general integral of an
equation and its family of subintegrals is exhibited in Art. 13,
in the equation
UmAP + BQ + CB+...
Now as these coefficients A, B, G,... are arbitrary and
independent it would appear on the face of this series that it
is incapable of being expressed in a finite form, there being
no law connecting the coefficients with one another. We have
seen however that by means of a germ and its powers we can
symbolically represent simultaneously in a finite form both
the integral itself and the arbitrariness and the independence
of the coefficients. There are cases in which this can be ac-
complished in more forms than one.
It is always an important object to assume a germ and its
powers in such forms and with such a law of coefficients as may
enable us to sum the series which is constituted of subintegrals
in a finite form. This cannot always be done; and when it
cannot, then the substitution of a germ and its powers for
Af B, C,... is generally useless; and recourse must be had
to the use of other means.
To make our meaning in this matter quite clear, we will
produce an example (of a very simple kind) of the process of
gathering up a whole family of subintegrals into one finite
form which contains them all, and yet at the same time im-
plicitly preserves the individuality and the independence of
every member of the family. This is the chiefest of all the
properties of a germ, and renders the doctrine of germs of
much importance. Nothing can well exceed their utility in
the discovery of symbolical equivalences ; and in the trans-
formation of integrals by means of these equivalences.
12 SOME GENERAL PROPERTIES OF GERMS.
17. Let it be granted that the following quantities con-
stitute as a whole a complete family of subintegrals, belonging
to a certain linear differential equation :
i. 5. iL + y. *L+wl. x* i *fy , y* . &<>
*' 1' 1.2 1' 3r 1.1' 4r l^.l"1-!^'
"We combine them on the principle of superposition into
a single integral U by means of arbitrary constants thus,
^t^+^+B^ffi+a
^g^M^
?2y . jf
1.2.1^1 .2,
in which A, B, C, ... are absolutely arbitrary and independent.
We now assume an extemporised germ m, and use it and
its powers to replace (or eliminate) the arbitrary constants
A,B,G,...
U-
= ^{l + m|
+ |)W(;
!!fl.li + ' J
= A(l + ^
+ 1.2
-}(*♦¥
+ 1.2 +
■)
= J$Gmx+m2V
• (I)-
We have here prefixed the common coefficient A, because
every integral of a linear equation takes an arbitrary general
coefficient. In this simple form of the complete integral are
contained (without loss of their individual independence) all
the members of the subintegral family because m is a germ.
We may also now obtain the differential equation of which
the above are the independent subintegrals. For by dif-
ferentiating equation (1) with respect to its independent
variables x, y, we find
d?U__dU
dx* ~ dy { h
Now in reference to the form of this differential equation
we may remark ; that it would not be changed or in any way
affected were we to write x + g and y + h for x and y ; neither
SOME GENERAL PROPERTIES OF GERMS. 13
would its form be changed by writing mx and m2y for x and y.
We shall express this by saying that this equation allows its
independent variables to take both major and minor germs.
In the above integral (1) the major germ m enters ex-
plicitly ; but if we write x + g, y + h for x, y in it, it takes the
following form,
^46m {x+g)+im? (y+h) — j^emg+mPh ^ €mx+miy — ^€mx+m2y^
Hence the exponential form of the integral (1) can be in
no way affected in generality by the introduction of major and
minor germs, the former being already present in it explicitly ;
and the latter implicitly, inasmuch as they may be said to lie
hidden in the external arbitrary coefficient A.
18. If one of the independent variables of a proposed
linear differential equation can take a minor germ, the family
of subintegrals can by means of that germ be cast in such
a form that all the subintegrals can be obtained from any
one of the family by simple integration and differentiation
with respect to that variable. (See Art. 4.)
Let ot . u — 0 be a linear differential equation which allows
one of its independent variables (as x) to take a minor germ g.
Then as the writing of x + g for # in ct . u = 0 produces no
change, we may do the same in any integral of ijt . u = 0 without
destroying it as an integral ; and as the substitution of x + g
for x in an integral would introduce the new germ g the gene-
rality of the integral would not be diminished; but on the
contrary it would be increased, unless the integral in which
the substitution is made be itself perfectly general. Hence
the general integral must of necessity be of such a form that
the substitution of x + g for x in it cannot affect its perfect
generality,
.-. u = F(x+g,y, *,...).
Let this be expanded by Taylor's Theorem in powers of g;
/, g d a* d2 g3 dz \ ^,
14 SOME GENERAL PROPERTIES OF GERMS.
The last step is by Art. 13; and A, B, G, ... are arbitrary
constants.
Hence A, B, G, ... are the coefficients of the subintegrals,
which therefore are in their order as follows [we denominate
F(x, y, ...) the first subintegral],
Ffay,.,.); sffo*-...)s fipFfay,...)! &c.
If any one of this family become known, then the whole
family may be found from that one, by integration and by
differentiation with regard to x.
From this property of minor germs, it becomes a matter
of no little importance, whenever it can be done, to reduce
a proposed equation which does not allow any of its independ-
ent variables to take a minor germ to a form that will allow a
minor germ.
19. The potentiality of a minor germ may always be repre-
sented in an equivalent form by means of an arbitrary function ;
that function being, however, not one of quantitative symbols
but of symbols of operation.
For when one of the independent variables (as x) takes a
minor germ (as g) we have seen that
u = F(x + g,y, ...)
If another variable (as y) take an independent minor germ
h, we should find in the same way that
d d^
dyj
and so on to any number of variables taking independent minor
germs.
=*(i'£)'*(-*:*r")-
SOME GENERAL PROPERTIES OF GERMS. 15
20. If an independent variable (as x) takes a minor germ
g, the general integral of the equation will always admit of
perfect symbolical expression in the form of an infinite series
according to positive integer powers of that variable.
For in this case
u = F(x + g,y, ...)
= F(g+x,y>...)
which is a series that contains x in positive integer powers only ;
and this series is symbolically the complete general integral by
hypothesis.
If two of the independent variables (x, y) take independent
minor germs, then the general integral will always admit of
being expressed in the form of an infinite series containing both
x and y in positive integer powers only.
For in this case
u = F(x + g,y + h,z ...)
= F(g + x, h + y,z ...),
from which the proposition follows by expanding F by Taylor's
Theorem.
It is evident the proposition may be extended to all the
independent variables that take independent minor germs.
21. If -sr f -7-,-7-J w = 0 be understood to represent any
linear differential equation of two independent variables (x} y)
with constant coefficients, then will each of these variables take
an independent minor germ; and consequently the complete
general integral of the equation may be fully expressed in any
one of the three following equivalent forms,
16 SOME GENERAL PROPERTIES OF GERMS,
or (ii) ...U = P + Q^+R^+...
in which P, Q, R, ... represent series of the general form
A+B*1 + C^ + ...
or (ni)...u=P+Q?+R^-2 + ...
in which P, Q, B, ... represent series of the general form
When hereafter we assume the first of these forms as repre-
sentative of the complete general integral of a proposed differen-
tial equation, we must remember that the sole authority for the
truth of this assumption lies in the fact that we know from the
form of the differential equation itself that both so and y take
independent minor germs. The second form supposes that y
takes a minor germ ; and the third form supposes % to take a
minor germ.
22. Let iff lx, j- , -=-) u = 0, or briefly ct . u = 0 denote a
linear differential equation in which one of the two inde-
pendent variables (as t) occurs only in the form of a differential
symbol of operation.
In this case the general integral is completely represented
by the following form,
u = F(se, t + g),
g being a minor germ of t
Now if we integrate such a differential equation as the one
before us by the method of infinite series, it will sometimes
happen that we shall obtain a result which may be represented
by the following :
in which A, B, C, ... are independent arbitrary constants; and
SOME GENERAL PROPERTIES OF GERMS. 17
they are therefore the coefficients of the members of the family
of subintegrals of which u is constituted.
Hence we are at liberty to eliminate these arbitrary con-
stants by means of the powers of a germ.
t f
Now A + B =■ + C z — - + ... is a MacLaurin's infinite series
X I . L
(see Art. 10) and its symbolical equivalent representative will
be under a certain restriction of form corresponding to this fact
(Art. 11); and the proper form may be found in the following
manner.
Equating the two preceding forms of u we have the following
symbolical equivalence :
F(x,t+g) = 1r(x,^(A+B{ + C^ + ...y,
and the left-hand member being a function of t + g, the right-
hand member must be so likewise ;
•'• A + B I + ° A + - =f(f+9) (!)•
Now /(£+#) =f(g + t), and whichever of these two forms
we adopt the result of the expansion of it in a series by Taylor's
Theorem must be symbolically equivalent to the left-hand
member of equation (1).
Hence
A + Bt- + C^-2+...=f(g)+f(g).tJ+f"(g).^2+...(2).
But in order that (2) may be a symbolical equivalent of the
series on the left-hand, /(#),/' {g),f (g), ... must be different
powers of the germ g ; a condition that requires the following
supposition,
where p must be of such a value as shall render the expansion
°f (9 + t)p an infinite series. Hence p must not be a positive
integer.
... A + B j + G^-2 + ... = (g + t)p, or = (* + gf.
18 SOME GENERAL PROPERTIES OF GERMS.
Hence u = yjr (x, £) . (g + t)p (3),
or »^(«, |) . (f +g? (4).
There are therefore two forms in which u may be presented,
a circumstance which is notable for the following reason.
If p were a positive integer (which it cannot be) these two
forms of u would be identical, because in that case the expan-
sions of (g + t)p and (t 4- g)p would be identical, with the exception
only that their respective terms would be in a reverse order.
But as p is not a positive integer the expansions of (g -f t/ and
(t+g)p are dissimilar, though symbolically equivalent (see Chap.
in.).
Hence that the integrals marked (3) and (4) are dissimilar
though symbolically equivalent, is due to the circumstance that p
is not a positive integer; and further that (t+g)p and (g + t)p
are symbolically equivalent.
Consequently we have two dissimilar though symbolically
equivalent forms in which we may finally present the general
integral of the proposed equation, whenever that integral can be
found in the form
23. If we expand (g + t)p in order to eliminate the germ g
and obtain the family of subintegrals of which u is constituted,
we obtain
the first subintegral = \jr (x, -7 -J . A = i|r {x, 0).
The other subintegrals may all be obtained from this by inte-
grating it with respect to t successively (Art. 4 contains an
example of this).
But if we expand (t + g)p the first subintegral will be equal
to i/r (x, -7- ) ,tp; and the other members of the family will be
SOME GENERAL PROPERTIES OF GERMS. 19
obtained from this by successive differentiations with respect
to*.
As it is always possible to differentiate, and not always pos-
sible to integrate, a given function of t, there will be an advan-
tage in using the form
first subintegral = yfr (at, j\ . tp (1).
But here crops up the question, — how are we to assign a
proper value to p ? for the preceding Article tells us nothing
respecting it but that it is not to be a positive integer. It does
not even tell us distinctly whether zero is to be classed among
positive or among negative integers. There is however no diffi-
culty in seeing that we must assign to p as small a value (apart
from its algebraic sign) as possible.
In some degree p is therefore a disposable numerical quan-
tity;* and we shall follow the rule of assigning to it the least
value (apart from algebraic sign) that will enable us to express
the first subintegral (1) in finite terms, for our object is to find
the integral of a proposed equation in finite terms.
24. One possible case must here be noticed. There being
nothing to fix a definite value of p in the investigation of
Art. 22, in the formula
first subintegral = yjr ix, -?-) tp,
if it should ever happen that this leads to a general subintegral
of the form W . cop, without the necessity of our assigning to p
any definite value, then p may be considered to be a germ, and
the first subintegral will be inclusive of the whole family of
subintegrals.
In this particular case therefore we have (by Art. 11),
u = W. cop
= W. <£(*>).
25. We have said that the smallest possible value (apart
from algebraic sign) must be assigned to p in order to obtain
2—2
20 SOME GENERAL PROPERTIES OF GERMS.
the first of the family of suhintegrals, and that from the sub-
integral so found all the other members of the family may be
obtained by differentiation with -^ .
J dt
From this it is obvious that if we can obtain a finite sub-
integral by assigning to p a value which is not the least pos-
sible (apart from sign) the subintegral so obtained will be one
of the family of subintegrals ; and we may ascend from it to
the first subintegral by successive integrations with regard to t
We shall know when we have arrived at the first by the circum-
stance that we have arrived at a subintegral which is not inte-
grate in finite terms.
26. The general principle that we shall adopt in the inte-
gration of linear differential equations is that of taking advan-
tage of any peculiarity that may be perceived to exist in their
forms, favoring the introduction of germs into their integrals;
for as an integral that is perfectly general cannot be made more
general, the introduction of a germ, though it may affect the
generality of an integral that is not perfectly general, cannot
make it less general ; but on the contrary every germ intro-
duced brings it one step nearer to perfect generality.
When therefore a differential equation is proposed for inte-
gration we begin by changing (if necessary) the dependent and
independent variables (see Chap. IV.) with the object of bring-
ing the equation to its simplest form, or to a form which will
enable us to detect the possible existence of germs in the
integral.
The preceding Articles will have made it evident that it
would be a great point gained if the reduction and transfor-
mation can be carried on till we have arrived at a form in
which one at least of the independent variables shall occur
only in the form of a differential symbol of operation, for such
a variable will take a minor germ. The following will illus-
trate the method of proceeding with such an equation, and will
also be useful for reference.
SOME GENERAL PROPERTIES OF GERMS. 21
27. To integrate ~ = vr (x, y, . . . ^ , g- , . . .J , or simply
= vs . u.
This equation allows t to take a minor germ, and therefore
(Art. 21) the following will be a perfectly general form of its
integral :
in which P, Q,B, ... are functions not of t but of the independ-
ent variables that occur in the operating function ts.
Substitute this form of u in the proposed equation
du
t /2 /3 \
1 ' 1.2 1.2.3
28. To integrate -p = ot (#, y, . . . -r- , t , . . . ) m, or briefly,
By the same method as the above, and with the addi-
tional consideration that this equation allows us to write in
any integral jt instead of t, we obtain the following general
form of integral :
•-(i+1k-+5«'+.--:.)*
in which P, Q are independent functions of the variables con-
tained in ot.
The two serial members of u are independent subgeneral
integrals; and their independence is due to the circum-
stance that -57 occurs in the differential equation only in the
form (-77) , and their independence is secured symbolically
22 SOME GENERAL PROPERTIES OF GERMS.
by the existence of the ambiguous symbol j in the latter
of them.
It will be noticed that one of the subgeneral integrals con-
tains even powers of t only, and the other odd powers only.
But we are at liberty to construct out of them, by addition
and subtraction, two other equivalent subsequent ] integrals,
each of which shall contain both the odd and the even
powers of t
The following form of differential equation, though it
belongs to the case of two independent variables only, will
be found important, for many equations that occur in physical
enquiries can be made to depend upon it.
29. To integrate </> f-r-J u = vr f a, -j-)u.
By the usual method of integration by series the inte-
gral of this equation can generally be obtained in a form
equivalent to the following:
in which T is an arbitrary function of t.
Now since the proposed equation allows t to take a
minor germ,
in which A, B, C, ... are the arbitrary coefficients of the mem-
bers of the family of subintegrals which constitute u. Their
places may therefore (Art. 13) be supplied by the powers of an
extemporized germ m ;
= emtyjr (a, m) = emtX.
SOME GENERAL PROPERTIES OF GERMS. 23
When X is found by the substitution of this value of u in
the proposed equation, then u is known from the equation,
u = emtX.
30. Let ct(-7-,-7-,-t-,...]m=0 be a linear partial dif-
ferential equation of any number of independent variables and
having constant coefficients ; then will
be an integral of it ; L, M, Nt . . . being germs subject only to
the following equation of condition,
0 = v(L,MtNt ...).
For
^Wx' dy' S'^V^T^W Ty> d^,'")6LX+My+
= eLx+3fy+^+- * (L, M, N, ...)•
Now as L, M, JV, . . . are germs, we are at liberty to assume
such a relation to exist among: them as will render the right-
hand member of this equation equal to zero ; and the only
condition necessary for that purpose is «r (L, My N, ...) = 0.
Hence subject to this condition U represents a quantity which
satisfies the proposed equation.
We have not said that U is the general integral of the
equation ; but as it contains independent germs it needs must
be one of a great degree of generality. As a matter of fact it
fails to be the general integral in such cases only as are dis-
tinguished by the recurrence of one or more of the operative
factorials into which -crf-^-, -j- , -7-,...) can in some cases
be resolved.
We shall be careful to prove the perfect generality of U in
every case in which we shall use it ; and then only shall we cite
it as the general exponential integral.
31. The germs L, M, N, ... being of the nature of general
germs are liable to contain imaginary quantities ; it will some-
24 SOME GENERAL PROPERTIES OF GERMS.
times be desirable to express the general exponential integral
in terms of real germs only. Let us therefore assume their
forms to be
L + il, M+im, N+in,
in which L, M, K, ... I, m, n, ... are all real quantities.
By this change the exponential integral takes the following
form
U=AeK+iI=AeKcosI+BeEsmI
= AeK cos (I + B) ;
in which K= Lx + My + Nz + ...
and I=lx + my + nz+ ...
and the germs L, M, N, ... I, m,n, ... are subject to the two
equations of condition into which the following equation neces-
sarily divides itself,
0 = <G7 {L + tl, M + im} N + in, ...).
32. If c be a germ contained in an integral TJ of a linear
differential equation tsr . u = 0, containing any number of inde-
pendent variables and its coefficients not being necessarily con-
stant; then will -r-, -rj, -tj > ... and generally </> f-r-J Ube
integrals of vr . TJ— 0.
The function (f> is conditioned by the equation <j> ( -v- J 0 = 0.
Now c being a germ is not contained in ct ; and therefore c
and ot are commutative symbols. Also w . TJ— 0.
Hence </> ( -r ) TJ is an integral of -sr . u = 0.
33. The following indicates the possible existence of quasi-
minor germs in some cases.
SOME GENERAL PROPERTIES OF GERMS. 25
Suppose we have before us a linear equation of the following
form,
' / d d d j / 7/ , \
V = ™[7r>J->'T>ax+<)y + cz> ax + oy + cz),'
in which a, 6, c, a, b\ c' are definite constants.
It is evident that we may, without affecting this equation at
all, write x + g, y + h, z + k for x, y, z respectively, provided the
values of g, h, k are restricted by the two following conditions,
0—ag + bh + ck, and 0 = ag + Vh + c'k.
Now as g, h, k are subject to these two linear conditions
only, each of them may be described as a definite multiple of
some one indefinite quantity I, which we may designate an
independent germ. This independent germ will be divided
among the three independent variables in certain definite pro-
portions, and be to each of them a minor germ, or rather a
quasi-minor germ ; for we have defined a minor germ (Art. 8)
as belonging exclusively to an individual independent variable.
Major Germs and Homogeneity.
34. By means of major germs we may extend the usual
definition of homogeneity in the following manner.
If a mathematical expression F(x, y, z, ...) be of such a form
that when max} mPy, myz, . . . are written in it for xy y, z ... the
germ m becomes a mere factor or coefficient of the whole ; i.e. if
the following form of expression holds good,
F (max, mfiy, m^z, . . .) = mp F(x, y, z, . . .),
in which a, /3, <y ... have definite values; then we say that
F(x, y, z, ...) is a homogeneous expression of jp dimensions.
We may also say that x, y, z, ... are respectively of the
dimensions a, ft, 7, ... and we shall speak of m as being a major
germ in this case.
The following proposition will be found very important in
future operations.
35. Every homogeneous linear partial differential equation,
whether its coefficients be, or be not, constant, will have all its
26 SOME GENERAL PROPERTIES OF GERMS.
subintegrals (that are due to the elimination of a major germ)
homogeneous according to the above definition ; and they will
all be of different dimensions.
Let u — F(x}y,z,...) be the integral of a homogeneous
differential equation. Then since a general integral is not
affected as to its generality by any change of the independent
variables which does not affect the differential equation, we may
write max, mPy, myz, ... in both the equation and its integral
without affecting them ;
.*. u — F (max, mPy, m^z, ...) (1),
and by differentiation of this we obtain the following equation,
/ d , 0 d d \ du /_>
l«*s+*j$+*J6+"-v*"" »as (2)-
Now expand the right-hand member of equation (1) in powers
of m\
.'. u = Pmp+Qm9 + Rmr + (3),
in which P, Q, R ... are functions of x, y, z ... but not of m;
they in fact constitute the family of subintegrals due to the
elimination of the germ m.
Hence each of them (i.e. of P,Q, R ...) is an integral of the
proposed homogeneous equation ; and consequently each term
of (3) will satisfy the equation (2).
Taking the first term Pmp and substituting it in (2) we find
dP^Q dP , dP A „
aXdx-+^d^+^zTz + '"=PP &>
the meaning of which equation is, that the subintegral P is
homogeneous and of p dimensions.
In the same way we learn that Q, R, ... are homogeneous
subintegrals of q, r, ... dimensions respectively.
The members of the family of subintegrals obtained by the
elimination of m have therefore this common property, — they are
all homogeneous ; but being of different dimensions their sum,
i.e. the general integral which contains them all, is not homo-
geneous.
SOME GENERAL PROPERTIES OF GERMS. 27
Homogeneity is, therefore, the distinctive feature of a sub-
integral.
36. As a major germ generally (though not always) belongs
to at least two independent variables, if a proposed differential
equation contains more than two such variables it may admit of
more than one independent major germ ; or if it admits of one
only, there may then be some independent variables that do not
take a major germ at all.
Hence it may happen that a differential equation may be
homogeneous with regard to only a portion of its independent
variables : and being homogeneous it may be of different dimen-
sions in reference to its different major germs.
Thus in the equation ^— j- = — , we may write mx, my, mH
for xt y, t respectively, and the equation is therefore homo-
geneous.
Or we may write nx, nt for x and t, and consider the equa-
tion homogeneous with respect to x and t. So it is homogeneous
with respect to y and t And we may write Ix for x, and t~xy for
y. But all these results are included when we write hnx for x,
rlny for y, and mnt for t, there being three germs involved in this
case. This is therefore the most general assumption of major
germs ; and it implies that the equation is independently homo-
geneous with regard to x, y, f ; and to x, t ; and y, t. It there-
fore possesses a triple homogeneity; and to obtain general
results all three must be taken account of.
It will now be manifest that the existence of major and
minor germs can oftentimes be discovered prior to integration
from the form of the proposed differential equation by mere
inspection. We shall see hereafter, however, that there may be
possible major germs which are not so easily discovered.
And it is always to be remembered that we are at liberty to
introduce into a known integral any possible germs, and that
the result will be still an integral of the proposed equation,
which may be thereby rendered one of increased generality.
CHAPTER III.
ON SYMBOLICAL EQUIVALENCE.
37. We consider the elementary quantities and magnitudes
with which we have to do as being measurable by numbers;
and an essential property of every such quantity or magnitude
is, that " the whole is greater than a part of it."
Zero, which is usually denoted by the symbol 0, we consider
to be " the negation of quantity or magnitude." The absence or
negation of a quantity cannot be divided into parts ; and what
has no existence cannot be treated as having properties.
But zero, though non-existent as a measurable quantity,
admits of symbolical representation by means of real quantities
in an infinite variety of ways ; as for example,
0 - x — x\ 0 = 1+ cos 2^—2 cos2a; ;
These are called equations, but we here speak of their right-
hand members as the symbolical equivalents of zero ; and hence
the mathematical sign (=) is to be understood not as always
denoting numerical equality, since zero is not a number, but as
(in such cases as these) denoting symbolical equivalence.
du nil
Also such a question as this, — find the integral of -*■ - + -j- = 0,
may be enunciated in the following equivalent form, — find the
most general form of u in terms of x, y which will render the
following equation a symbolical equivalence I -j- + -=- J u = 0.
ON SYMBOLICAL EQUIVALENCE. 29
The reader will kindly keep in mind, whenever he finds the
sign (=) connecting two quantities, or two steps in an investiga-
tion, which are not equal algebraically, that that sign is in this
case to be read as signifying symbolical or integral equivalence.
We might preserve at every step, and in every equation, both
algebraic and symbolic equivalence, but this would have to be
done oftentimes at an inconvenient expenditure of time and
new algbraic symbols. After a little practice no inconvenience
will be found in the system employed in these investigations
which are chiefly about integrals. The sign ( = ) has three
meanings : — algebraic equality ; — symbolical equivalence ; — and
equal in generality as integrals.
38. For a reason analogous to that which leads us to
reject zero as a numerical quantity we reject infinity, for it
cannot be numerically increased by addition nor diminished
by subtraction, since it is not measurable by numbers.
Nevertheless there is a case of infinitude which can be dealt
with to advantage, viz., the case of series the number of whose
terms is infinite.
Infinite series are of two kind* : —
1. A series may have a first term but no last term; or, in
other words, it may have a beginning but no end.
2. A series may have neither a first term nor a last term.
1 + 2 + 22 + 23 + . . . ad infin. is an example of the former
kind, and ... + ^ + ^ + ^ + 1 + 2 + 22 + 23+ ... of the latter
kind.
39. The meaning of the word equivalence which it will be
necessary to attach to the sign ( = ) in some of the subsequent
articles is so unusual that we shall add a few more illustrations.
The late Professor De Morgan proposed the following
equation for solution,
x = 2x.
If x be in this equation a numeral quantity, divide both
sides of the equation by x. Then on the ground that if equals
30 ON SYMBOLICAL EQUIVALENCE.
be divided by equals the quotients are equal, we find 1 = 2,
a result which we are obliged to reject though obtained ac-
cording to the acknowledged principles of numerical reasoning.
Hence the only remaining inference is that the equation
before us is not one of numerical magnitudes. The equation
when put in a verbal form is this, " Find a numerical magnitude
that shall be numerically equal to the double of itself." When
thus stated the equation is seen at once to involve of necessity
a property irreconcilable with the properties of numerical
magnitudes.
40. But it remains to be ascertained whether the equa-
tion x = 2x admits of a solution reconcilable with symbolical
equivalence. Find x so that it shall be symbolically equivalent
to 2x, is now the problem before us.
There is no particular difficulty in finding the following
answer to this question,
a = ^(...+J + i + J + l + 2 + 4 + 8+...),
where A is an arbitrary constant.
Hence the right-hand member of this equation is a sym-
bolical equivalent of zero, which is all that is meant by the
equation
0 = J.(...+i + l + J + 1 + 2 + 4 + 8 + ...).
41. Let it be required to find x such that it shall exceed
its double by unity.
The algebraic equation for this case is
x = 2^ + 1,
of which there are three solutions, viz.
x = — 1,
#=1 + 2 + 4 + 8 + ...
and * = - (4 + 1+4+—)-
The first and last of these may be numerically equal ; but
it is evident that the second being a positive quantity cannot
ON SYMBOLICAL EQUIVALENCE. 31
be numerically equal to either of the others. Hence the fol-
lowing are nothing but symbolical equivalences,
-1 = 1 + 2 + 4 + 8 + ...
i + i + i + ... = -(l + 2 + 4 + 8+...) (1). •
Both the terms of this last equation are legitimate expan-
sions of the same symbolical expression, — - .
It is only in reference to the problem algebraically ex-
pressed by the equation x = 2x + 1, that we maintain these
equivalences to be real. They all satisfy this equation; yet
they are not its roots but its equivalences.
Another important matter is that the equivalence marked
(1) shews that two infinite series may be strictly equivalent
though one of them may be convergent, and the other di-
vergent.
42. We come now to speak of another matter which we
shall denominate "integral-equivalence" as being distinct from
algebraic equality. Brevity of expression is the chief object
to be attained by the use of this kind of equivalence. An
example will best explain its nature.
In integrating the equation -^— 2=u by the method of infinite
series we find
7/
^(l + I^ + ....) + 5(f + TTJ-3 + ....),
in which the arbitrary constants A, B indicate that the entire
integral u consists of the sum of two independent integrals,
which we denominate subgeneral integrals. Each of these
is a perfect integral in itself and expressive of properties or
relations peculiar to itself. One of them contains only odd
powers and the other only even powers of x.
Having found the integral of the proposed equation in the
above serial form we proceed to introduce integral equivalences
32 ON SYMBOLICAL EQUIVALENCE.
in the following manner, with the object of presenting the
integral in the briefest possible form.
= Aex + Be~x
= AeJx.
This is our final result, and simple as it is, it is perfectly equi-
valent as an integral to the two infinite series which constitute
the entire value of u. In deducing it from those series the
arbitrary constants have suffered changes of identity at every
step, but we have been careful to preserve their only essential
quality, that they are arbitrary constants all through the
process of reduction.
Had the equation to be integrated been -j-^ + w = 0, our
result would have been
u = Ae1x.
It will be seen from the above example that we shall
hereafter feel at liberty to use the sign ( = ), as denoting in-
tegral equivalence ; and that in so using it we shall consider
not the identity of the quantities denoted by A, B, C, ..., but
merely take care that each shall preserve its only essential
quality, viz., that it denotes a perfectly arbitrary and inde-
pendent quantity.
43. If / (x) be expanded in an infinite series it is usual to
represent the result thus,
f (x) = Ax* + BxP + Cxy + ... ad inf.
Professor De Morgan proposed that the left-hand member
should be denominated the Invelopment of the right-hand
member. We shall adopt this designation. It has been usual
to speak of f(x) as the sum of the series, but unless the series
be convergent this designation is incorrect.
When we meet with two symbolically equivalent series,
if we can find the invelopment of one of them we shall use that
ON SYMBOLICAL EQUIVALENCE. 33
series in preference to the other without reference to its con-
vergence or non-convergence.
44. A series that has no last term may possess properties
not possessed by the sum of any number of its terms. Take
the following example :
U=l-x + x2-x*+...ad-'mL (1).
If we multiply this series by x and subtract the product
from unity it remains unchanged ; and this is not a property
of the series continued to n terms only. Hence the following
is true of the infinite series only, viz.
U=l-xU;
1
U=
\ + x'
This is the invelopment of the series; i.e. it represents the
whole infinite series, and if it be expanded according to the
usual rules it will be found to produce the whole series.
But = -, and the latter expression is the invelop-
-L "t~ X X -J- JL
ment of the following infinite series,
= -* + -s--4+... ad inf. (2).
x + 1 x ar xs x4,
Now since the invelopments of these respective series are
symbolically and algebraically equal, we say that the following
is both a symbolical and an integral equivalence,
1 — x + x2 — . . . ad inf. = T + - 3 — ... ad inf. ;
and therefore in reducing an integral to its simplest or most
manageable form we should not hesitate, if necessary to secure
ultimate success, to introduce this equivalence, or any other
which rests on the same basis.
45. We shall now generalize the above results by shewing
that the two following infinite series are symbolically equivalent,
A + Bx + CV + ... « A + Bx~x + Cx'* + ...
E. 3
34 GN SYMBOLICAL EQUIVALENCE.
in which A, B, C, ... are constant quantities, definite or indefi-
nite.
Instead of C, B, E, ... in the left-hand member write
respectively C - 2B, B' - 4 C + 3J5, E' - 6B' + IOC" - 4£, &c.
.*. A+Bx+Cx2 + Bx*+...=A+B(x-2x* + 3xs- 4^4 +...)
+ C'(a2-4.z3 + l(k4-...)
+ B' (xs- 6x* +...)
+ &c.
Bx C'x2 B'x*
{l+xf ' (1 + *)4 (l+#)6 '
. igar1 (7Va DV8
= A + Bx-1 + Cx-^ + Dx-* + ...
The validity of this investigation depends entirely on the series
being infinite, and it cannot hold good for n terms, with the
single exception of n = 1.
It is to be noticed also that the quantities G\ B', E\ ... are
used in the proof merely as artificial means of distributing the
terms of the left-hand series into groups suitable for our purpose.
And it is obvious that a different grouping would have led us to
another type of symbolical equivalence; as will be seen in the
following Article.
46. To shew that the two following infinite series are sym-
bolically equivalent to each other ;
A + Bx + Cx2 4- ... = xp (A+Bx-1 + Cx~* + ...),
the index p being subject to the sole condition that it must not
be a positive integer.
Instead of B, C, B, ... in the left-hand series substitute the
following quantities,
X I . —
ON SYMBOLICAL EQUIVALENCE. 35
1 I . — [._..)
&C. = &C„
the law of these substitutions being obvious, and requiring, as
the series are infinite, that p shall not be a positive integer.
On making these substitutions and proceeding step by step
as in the preceding Article we arrive at the following symbolical
equivalence,
A + Bx + Cx* + ... = xp (A + Bx*+Cx* + ...)
= Azp + Bxp-l + Cxp-*+...
47. This result may be presented in the following form,
"'©•
And if A, B, C, . . . are absolutely arbitrary and independent,
then is also the function F( -=-) an arbitrary function of -j- ,
subject only to F( j-\ 0=0.
The value of this result in the discovery of subintegrals will
be seen when we come to the actual integration of equations.
The reader may compare these results with Art. 23.
3—2
CHAPTER IV.
THE TRANSFORMATION OF LINEAR DIFFERENTIAL EQUATIONS
OF THE SECOND ORDER.
I. Two independent variables; coefficients constant.
48. Our object in this chapter is to reduce equations to
their most simple forms with thewiew of discovering those forms
which present peculiar integrational difficulties.
We may classify any linear differential equation of two inde-
pendent variables and having constant coefficients under some
one of the four following heads,
/ x « d2a A du „ du ~
x ' dxdy ax dy
. ~ ~ d*u , , x d*u 7 d?u A du ^ du „
(^...^1?+{a + b)1^dy + abw + A-d- + BTy+Cu.
, x A / d . ad\2 . .du T,du t n
*«* r, ( d , ad\* . f d , ad\ „
49. To reduce the form (a) assume a new dependent varia-
ble v such that
n=v€-Ay~Bx.
This gives the following reduced form when substituted for u,
TRANSFORMATION OF LINEAR DIFFERENTIAL EQUATIONS, &C. 37
which comprehends the two following elementary forms,
/iN ■ d2co , d?co . .
W 0=dXdy'mdd^ry='a <2)-
The former of these presents no integrational difficulty; and
the latter we shall integrate in a future Article.
50. To reduce the form (£) we change the dependent
variables by assuming two new variables f, rj such that
x=z^+rj and y = af + brj.
du _f d d\ , du_/dhd\
dg \dx dy) ' drj \dx dy) *
and the reduced equation is
d2u B — bAdu B — aAdu „
af drj a — bdl; b — a drj
which being of the form (a) can be reduced to the forms (1) and
(2), and therefore furnishes no new integrational difficulty.
51. To reduce form (7) we assume x = ^ + tj and y m af -f- -j rj ;
du (d d\ . du fd B d\
" -dr\TX + ady)U' and ^{dx + Afyh
and the following is the form of the reduced equation,
A d2u . du n
and by changing the dependent variable, if necessary, by writing
vemrl for u, m being such as to satisfy the equation Am + G = 0,
we obtain the following form,
The following are therefore the ultimate forms furnished by
form (7),
,Qv (Fa d?co da) . .
(3) 0 = «P + <°' c&=dj W«
and S = ° (5)>
of which (4) only presents any new integrational difficulty.
38 TRANSFORMATION OF LINEAR DIFFERENTIAL EQUATIONS
52. To reduce form (8) we assume x = f + rj and y = a^ + br)\
du ( d d\
and the reduced equation is
A d2u .du , n
0 = d? + AdS+Cu>
a form which does not contain rj I which is equal to — —j- J and
presents no integrational difficulty. It is in fact an equation of
one independent variable ; and consequently, when it is inte-
grated, arbitrary functions of rj, or rather of (ax — y) must be
used instead of arbitrary constants.
53. Hence gathering together the forms that are of difficult
integration we find only the two following,
d2u i d2u _ du
dx dy ~ dx2 dy '
These forms it will be our business to integrate in the follow-
ing chapter.
II. The case of three independent variables,
54 We may arrange any equation of this class under some
one of the four following heads,
, v ^ d2u Adu 7> du , ~ du , T^
(a)...0= 7— r +A-J- + BJ- + C j +Aw.
x ' dxdy dx dy dz
d\i a?u .du ^du ~du j~
W'" - dxdy dxdz " dx dy dz
, . rt d2u d2u t , d2u A du T? du ndu „
^■■■0 = d^j + aJ*j2 + bd^z + AT. + Bd1/ + C<h + K<>-
,., . dhi d*u , 0d'u , d°u . d*u d'u
(8). . .0 = jj, + a <7? + ,8 -, + a dxd;/ + b 2^+ 0 ^
dx dy dz
OF THE SECOND ORDER. 39
55. To reduce the form (a) we assume u=ve~Ay~Bx~mzi
where m is a constant that satisfies the equation mC+ AB=K,
and the following is the reduced form of the equation,
which includes the two following elementary forms,
n * ^*a) =o d -^2ft> = — (2)
* ' dxdy ' dxdy dz" r; ''
The former of these presents no integrational difficulty.
56. To reduce the form (fi) let x, y, £ be a new set of inde-
pendent variables, in which £=z — ay. The reduced form of
the equation is
d2u Adu ^du ir* „. du T,
dx dy dx dy d£
which coinciding with form (a) introduces no additional integra-
tional difficulty.
57. To reduce form (7), for u write vemx+nl,+re, the constants
m, n, p being such as will satisfy the three following equations :
0 = A + n + ap, 0 = B+m + bp, and 0 = G + am + bn.
The reduced equation is the following,
d2v d2v 1 d2v rT,
dx dy dx dz dy dz
in which K' — Cp — mn -f K.
Let now the independent variables be changed to x, y, f
where f= z— ay — bx. By this means the reduced equation
becomes
<Pv , d2v , „,
dx dy d£2
which includes the two following new elementary forms,
/m d2co d2co , d2<o d*co ...
(3) -j — r- = -^-, and -y— j-=-To* + a> (4).
w dxdy d£2 dx dy d?
40 TRANSFORMATION OF LINEAR DIFFERENTIAL EQUATIONS, &C.
i
58. To reduce the form (S), assume a new set of independ-
ent variables f , rj, f such that
% = x — gy, t) = y — hz, and f = z — kx,
the constants g, h, k being such as will satisfy the following con-
ditional equations,
0=og*-ag+l, 0=/3h2-ch + l, and 0 = k2-bk + /3.
By these means the form (5) will be reduced to form (<y),
and consequently introduces no new elementary forms.
59. Gathering together the elementary forms which pre-
sent integrational difficulties, we find that they are the three
following :
d2u _ du d2u _d*u , d?u _ d2u
dx dy dz ' dx dy dz2 ' dx dy dz2
In this chapter we are therefore presented with five difficult
linear differential equations of the second order with constant
coefficients ; viz. two when there are two independent variables,
and three when there are three independent variables.
CHAPTER V.
INTEGRATION OF EQUATIONS OF TWO INDEPENDENT
VARIABLES.
In the preceding chapter we have seen that the two fol-
lowing equations present the only difficulties that are experi-
enced in the integration of linear equations of the second
order with constant coefficients. In this chapter we shall
bring in the properties of germs to our aid in the task of effect-
ing their complete integration.
60. To integrate 3^ = ^-
According to Art. 21 the following is a series which may be
assumed for the complete integral of this equation,
which being substituted in the proposed equation gives the
following complete form of u,
where P«4+JSj+ <^T2 + -j
the constants A, B, C, ... which are absolutely arbitrary, being
the coefficients of the subintegral constituents of u. Hence we
42 INTEGRATION OF EQUATIONS
may write M, M2, if3, ... for them (Art. 13), M being an extempo-
rized germ ; and then we shall have the following equivalence,
p
= A+B*\
^1-2 +
Ae
Mx .
y
u
d2 f
dx2 + 1.2'
d4
dx*
+ .
Thus it is proved, for the proposed equation, that the general
exponential integral of Art. 30 is the complete integral.
In this form of the general integral the minor germs of x and
t are implicitly contained in A, the general coefficient ; and M is
a general germ, i.e. it is liable to contain both real and imagi-
nary quantities. The major germ is explicitly contained in
the integral, on which account the integral takes a form which
we may refer to as the major-germ form.
61. In Art. 31 we have shewn the general method of ex-
pressing an exponential integral, that contains general germs, in
an equivalent integral containing real germs only.
In the integral just found we have merely to write M+ im
for M; and the following is the form in real germs M, m ;
.-. u = A^M2-m2»+M* cos m (23ft + x+B).
The minor germs of x and t are in this integral implicitly
contained in the arbitrary constants A, B. The form is a
major-germ form.
62. To find the integral of ^-r2 = -77 in a minor-germ form,
i.e. in a form which renders the major germ latent in the arbi-
trary constants of the integral.
From Art. 34 we learn that the subintegrals obtained by
the elimination of a major germ will all be homogeneous and of
different dimensions. This therefore suggests the following
method of obtaining the subintegrals required.
OF TWO INDEPENDENT VARIABLES. 43
Let V be a function of x and t which is of zero dimensions.
The general representative of such a function in the case of the
proposed example will be V — <f> f-r-J , for if M x, APt be written
x
in this for x, t, the germ M will disappear. Denote -j by v, and
then the following general form of homogeneity will represent
any one of the subintegrals,
P = t*V,
the dimensions of this subintegral being p.
This being an integral of the proposed equation must satisfy
it, and being substituted therein, the following is the resulting
equation for the determination of V,
dv% + 2 dv P
Now we wish to obtain subintegrals in a finite form, or if
that be not possible, then in a form that shall give a finite
expression for u.
We make use of p (which is disposable) for this purpose ;
and enquire what value of p will give a finite expression for V.
We can see at once that p — — \ will answer our purpose. The
above equation being integrated on this supposition, we find
t)2 v2 r v2
V=Ae~~i +Be~~z \e1dv\
.-. p = tpV= Ar*e~u + Bt^e'it e*dv.
The last term we reject because it is not in a finite form ;
„/d d\ ^i _*2
•'•U = F[dx>dt)'t2eU'
Now the proposed equation shews that -j- when applied to
any integral of the proposed equation is equivalent to (-r-)
applied to the same integral ;
44 INTEGKATION OF EQUATIONS
and this is the general integral of the proposed equation, in a
form that renders the major germ latent. (Here the sign =
denotes integral equivalence ; and F stands for the words " ar-
bitrary function of.")
Lest the reader should have any doubt of the generality of
this result we will obtain it in another manner.
d?w du
63. The proposed equation -z—2 = -^ has constant coeffici-
ents, consequently the following is by Art. 21 the general as-
sumption for its perfect integral,
in which P, Q, R, ... are serial functions of t of the general form
A+Bi+cJ^ + ...
The substitution of this value of u in the proposed equation
furnishes the following form of the general integral,
fa if d * a? \n
(!)•
These are the two subgeneral integrals ; and the former
contains only even powers of x, and the latter only its odd
powers ; and this is due to the fact that -j- occurs in the pro-
posed equation in the form (-=-) only.
OF TWO INDEPENDENT VARIABLES. 45
The first subgeneral
■Kit-)'
= * (J) (« + <?)* see Art. 22,
Our wish is to obtain the subgeneral integrals in a finite
form, and therefore we now ask what value of p will enable us
to find the iovelopment of this infinite series. There is no
particular difficulty in seeing that p = — J will enable us to
do this;
:. first subgeneral = e ~Ht+g) (t+g)~*
= F(^\.t-ie~ft (Art. 19).
And from this we can deduce the form of the second subgeneral
integral.
Differentiate with -=- .
ax
(X X9 d ' \ /„ n t ti t \
Now the right-hand member of this is of precisely the same
form and generality as the second subgeneral integral in (1) ;
.*. second subgeneral =f(ji)-j-t~ie*t
46 INTEGRATION OF EQUATIONS
with the understanding that this integral shall contain only odd
powers of x.
Hence if we gather the two subgeneral integrals together we
have two terms, of which one contains only even powers of x,
and the other only odd powers ;
■■—('(£♦/(£)} ..-..*
-'($■<-'•*.
which agrees with the result obtained in the previous Article.
64. Hence we have found the two following forms of the
general integral of the equation ^— 2 = -j- ,
(l)...u = Ae™t+Mx,
in which the major germ M is explicit, and the minor germs
latent; and
in which both the major and minor germs are latent in the
general operative function F ( -7- J ;
-m-
It will not be forgotten by the reader, that when (=) does not
denote algebraic equality, it denotes the words "symbolical
equivalence."
65. To integrate -^ — =- = u.
ax ay
Both x and y take minor germs. Hence the general integral
can be completely expressed in a series containing only positive
integer powers of x and y.
OF TWO INDEPENDENT VARIABLES. 47
The following may therefore be assumed as a general form
of u,
P, Q> R, ... being serial functions of x of the general form
Substitute this form of u in the proposed equation ; and the
following is the result (in which we take the liberty of using
dx
-t for the symbol of integration),
-t(»z)('+*i+«nt+'--)
- + (,*).,- Art. (29)
(1) = Aecx+C~^, c being a general germ.
Hence the exponential integral
u = AeMx+Ny, subject to MN =■ 1,
is perfectly general in the example before us in this Article.
We may obtain the first subintegral in the following manner
by the elimination of the germ c from (1).
The form of the proposed equation shews that the product
(xy) is of zero dimensions. Let v* = xy, and let Fbe a function
of v. We may assume the following as the general representa-
tive of subintegrals,
P = xpV.
This being substituted in the proposed equation gives the
following for the determination of F,
av v dv
48 INTEGRATION OF EQUATIONS
This will be integrable in a finite form if we assume 2p + 1 = 0,
and therefore p = — \ ;
.'. V = Ae2v + Be-2v
= Ae2iv,
and the first subintegral P = x~^V
— cc~le2J^*v -
■••— VtB'S-^^* »
Now it appears from the form of the proposed equation that
the symbolical product of -j- and -r- is equivalent to unity, when
applied to any integral of that equation ; the above form of u
may therefore be presented in the following equivalent form,
The following equation is obviously true, and it gives rise to
the latter of these two subgeneral integrals,
— . x'h2^ = y-ie2^**
dy J
It has therefore been proved above that the elimination
of the major germ c from the exponential integral
u = Aecx+c~l* (4)
gives the following form of the first subintegral, to which we
shall often have occasion to refer,
P^x-h2^,
Now the integral (4) is expressed in terms of c as a general
germ j but we may express it in real germs by writing
c (cos m + i sin m) for c,
and c-1 (cos m — i sin m) for c-1,
in which new forms of the germs, c and m are to be considered
real germs.
OF TWO INDEPENDENT VARIABLES. 49
Let K=cx + c~1y, and I—cx—c~ly, then the integral (4)
will take the following equivalent general form of expression in
real germs,
u = AeKco*m coa (Ismm + B) (5).
d?u
66. The various integrals of , , + u = 0 may be deduced
from the two preceding Articles by writing therein — y for y.
.-. u = AeM*+Xv = F(J^,^\.x~i cos (2 Jw + B)
subject to MN+ 1=0.
The following Article is introduced for the purpose of future
reference.
67. To change the independent variables of the expression
d2u
dxdy '
Let f and 77 the new independent variables be such that
d d d , d d , d
~j~ ~ ji- + a -J- i and -y- = -tt + b -v- .
dx dg drj dy d% drj
These assumptions require that a, b shall not be equal.
.*. g = x + y, and 77 = ax + by, ■
and also, x = - =? , and y = — v~ , .
a-b ' * a-b
&u =/d d \ (d_ .d\
dxdy \d% dy) \df; drj)
2? + (a + *>3f3* + aW (1)'
Also (a-byxy^-^-aQirj-bg)
= -(v*-a + bv% + ab?) (2).
68. To integrate
50 INTEGRATION OF EQUATIONS
We deduce the required integral from Art. 65 by writing
therein the above values of x and y in terms of f and rj.
^W-a+bnt+abp)*
69. To integrate the equation (-7-2— j-J t*=0, in which
-r~, j — -T-] is repeated w times.
Changing the dependent variable, assume either u = e?™* F,
We begin with the former (m being a general germ).
-.Afll+y)"-1,
A. being a minor germ of y.
... u-**» Y
= Ae>mx+m*v(h + y)n-1 (1).
Had we taken the form u = em*" X we should have found
=(£-w)"(i+m)"x'
OF TWO INDEPENDENT VARIABLES. 51
' which is equivalent to the following integrals,
(a— )"x=°- and (i+mJx-0'
.*. w = €mV{^l€^(^ + ^)n-i + Be~mx(x + l)n-1} (2),
which agrees with (1) ; since As?™* represents both Ae™ and
Be-™.
A similar method of treatment will succeed with the equa-
tion
\dx dy )
dy
We have hitherto confined ourselves to equations with
constant coefficients; but in the following examples the coef-
ficients are functions of one of the independent variables.
70. To integrate
d2u adu b du
dx dy xdx xdy'
In this equation x and y can take a major germ ; and y can
take a minor germ also.
Hence changing the dependent variable we assume the fol-
lowing general form for the integral of this equation,
u = emvX,
7n being the major germ, and X being a function of x.
This value of u being substituted in the proposed equation
gives the following for the determination of X :
(•-£>■£-«
= A (mx — a)*,
.-. u = A{mx-a)b<rv (1).
4—2
52 INTEGKATION OF EQUATIONS, &C.
In this integral the minor germ is latent, and the major
germ is explicitly involved in it. Also m is a general germ.
Again, to find the general integral in a form which renders
the major germ latent.
We assume afV as the general type of the subintegrals ;
7 being a function of v, and v = - .
x
Substituting of V for u in the proposed equation we find the
following equation for the determination of 7 in a finite form ;
d ( dV\ ., .dV , lr A
That this may be integrable immediately the following
condition must be satisfied ; ap = — a.
Consequently in the general case /> = — !; but when a = 0,
p will be subject to no condition. a=0 is therefore an ex-
ceptional case.
d ( dV\ , n , . N <27 Tr _
This equation being integrated, gives the following as the
first subintegral,
7 ^*6 ? .fe6 ?
p=JL = ^.6
r^6 ^ r
+ 3+r«" J «*«■"* (2),
x yb+l y
and u^F^\(x1V).
Thus the proposed equation is completely integrated in a
form that renders the major germ m latent; but the term
multiplied by B will not be in a finite form, and will therefore
have to be rejected, unless b be a positive integer.
We will now take the exceptional case, viz. when a = 0.
mm... , <Fu b du
71. To integrate -= — 7- = - -=- .
0 ax ay x ay
.-. u = xlF(i/)+f(y).
CHAPTER VI.
EQUATIONS NEARLY RELATED TO. LAPLACE'S EQUATION.
Coefficients constant.
72. To integrate gj-^.
Let M be a general germ.
= <l>(x+jt) (1).
Expressed in terms of real germs only, we have by the
method of Art. 31,
u = A<F(X+M cosm (x +jt + B) (2),
M and m being in this form independent real germs.
To obtain the integral from which major germs are elimi-
nated, let us assume v = -, and V=(j>{v). Then all the sub-
t
integrals will be of the form P = tpV, which substituted in the
proposed equation gives,
This equation will be integrable at once, and therefore
in finite terms, if we assume —2p = p(p — 1),
54 EQUATIONS NEARLY RELATED
The two roots of this are p — 0, and p = \. The former
is of a doubtful character as to whether zero is or is not to
be considered to be not a positive integer ; the latter we see is
allowable.
We try the former, rejecting that part of the result which
is not in finite terms, and find
F=P = ^logVr
= illog— * (3).
°x + t K '
To find the second subintegral, or rather the first sub-
integral corresponding to the second subgeneral integral, we
assume p — — 1.
' ' dv*
d ( 2dV\ dV
dv\ dv J dv
Av + B
V=
v*-l
and *-*r,*T£r W'
= A(x + t)-1 + B(x-t)-\
■■':r*(S*
= F(x + t)+f(x-t)y
which agrees with equation (1).
But as the value of P in (4) is of the dimension (-1) we
may by integrating it with regard to x raise it to the dimen-
sion zero; in which case the first subintegral will be
73. Change the independent variables of the equation
dx* ~ df;
TO LAPLACE'S EQUATION. 55
and let the new variables f, rj be such that
f-logJS^?, and „-log(J=D*,
then
d2u _ d*u
Hence the form of the proposed equation is not changed by
this change of variables ; from which it follows that we are at
liberty to write log Jx2 - t2 for x, and log f — 77^) for t m any
integral of
d2u _ d2u
dxz~~dfy
and the resulting formula will be an integral of the same equa-
tion.
74. We shall now consider the equation
d2u d2u _ ft
dx2 dy2
We begin with the following proposition respecting this
equation and its integral. If in an integral of the proposed
equation we write ax -hjby for x, and ay —jbx for y} the result-
ing formula will be an integral of the same equation; a, b being
arbitrary constants.
Let g = ax +jby, and t) — ay —jbx, and let f, tj be the new
independent variables. We find that the result of this change
is the following differential equation
d*u d2u _
df* + dv2~
Hence the form of the equation is not affected by this change
of the independent variables; and consequently we may write
the above values of f , tj instead of x, y in any known integral of
the proposed equation and the resulting formula will also be an
integral of it.
On this we may remark that the substitution of ax+jby
and ay— jbx for x and y will introduce two germs a, b into the
5G EQUATIONS NEARLY RELATED
integral; and if the integral in which this substitution is made
had been deficient in the number of germs it contained, the
integral that results from these substitutions will contain two
additional germs, and may possibly now contain the requisite
number to render the integral general.
An example will illustrate this.
U=ex cosy
is manifestly an integral of the proposed equation, and it con-
tains no germs. Make the above substitutions for x and y; then
the following is also an integral of the proposed equation, and it
contains two germs a, b,
Z7= eaz+ibv cos (ay -fix).
If into this we introduce the minor germs of x and y we
have the following result which is (as we shall presently prove)
the general integral of the proposed equation,
U= Aeax+^ cos (ay -jbx + B) (1).
7o. To integrate -^ + — = 0.
The general exponential integral is
U^AlJtix+iy) (ty
= (f> (x + iy\
(in which we are at liberty to write ax + jby for x and ay — jbx
lor y).
Let r2 = Xs + y2, and tan 0 = - , . *. x = r cos 0, y = rsin0,
x
. u — <f>(r. cos 6 + % sin 6)
= <t>(re»)
= ^(rel*fl) + i|r(r6-^)
= 0 (rhie)
— Arinein0
= (Arn + Br~n) (aen9 + be~nd)
= (Arn + Br~n)(acosn0+bsmn0) (2),
TO LAPLACE'S EQUATION. 57
n being a germ ; and A, B, a, b being independent arbitrary
constants.
76. Let r and 6 be made the independent variables instead
of x and y; then the equation of the preceding Article takes the
following form,
(rd\* , d*u n
from which we learn, that we may write in any integral of the
equation of the preceding Article ri for r, and j6 for 0.
Also 6 takes a minor germ, and r a major germ.
If we seek the first subintegral after the manner of Art. 72,
we find
first subintegral = A log Jx* + y1 + B tan"1 * .
x
■■■u=F&°zj^*+f{iy™'i »
The integrals (1) and (3) are symbolically equivalent.
If^log^ + 2/2,
Hence we may write log Jx2, + y2, tan"1 - for #, y in any in-
x
tegral of the equation of the preceding Article, and the resulting
formula will be an integral of the same.
►tit m • x d2u d*u
77. lo integrate -7-5 +-7-3 = u-
The general exponential integral may be presented in the
following form,
U = A^M +W x+(M-N)iy
— jleM(z+iy)+N(x-iy) ^
subject to the condition 4iMN= 1.
58 EQUATIONS NEARLY RELATED
This equation of condition will be satisfied if we assume
2M = c (cos m + i sin m)
2N = c_1 (cos m — i sin m)
in which c and m are real germs, and also independent.
Let K = J (x cos m — y sin m)
and I—^{y cos m + x sin m)
.-. w = ^e^+«r1)^cos{(c-c-1)/+JB] (2).
78. If we now consider c a general germ, and assume
2M =s c, and 2JV = c"1, we find the exponential integral in this
form,
This form of the exponential integral agrees, as to its germ c,
with equation (1) of Art. 65, from which we learn that the follow-
ing is the form of the first subintegral P,
P = (x + iy) ' * ej\/(x+iy)(x-iy)t
This comprehends the two forms, (r2 being equal to x* + y2),
P = {A (x+iy)~t + B (x- iy)"i\ eK
Now x + iy = r (cos 0 + $ sin 6) = re5®.
••• -(;
A 2 7? *\
= r-i(^62 +JB€"2)(a€'' + 5€-0 (1).
in which A, B, a, b are independent arbitrary constants.
This value is symbolically represented by the following brief
equivalent,
0
u = Ar-leir€2 (2).
79. The integral of j-2 + -j-% + u = 0 may be deduced from
the preceding Article by changing the algebraic sign of c"1 but
not that of c.
.\ u = Ae^-o-W cos {(c + c"1) /+ B],
TO LAPLACE'S EQUATION. 59
and P = (x + iy)~ * ^to+tyuy-x)
80. There are several important equations related to
Laplace's equation which are reducible to the following type,
d?u _ d?u a du
df da? x dx '
We shall denominate this, when a is a positive quantity, the
standard equation of this type, for a reason which will be seen
presently.
For a small number of particular values of a this equation
has been integrated, but for general values of a it has not been
integrated.
The value of a admits of reduction in the following manner.
Let a = 2n -f b, 2n being the greatest even integer in a ; and
let a) be an auxiliary dependent variable such that
d*(o _d?co b dco '
W~dxr'¥xdx~ ('
d f dco\ , t / d(o\
~ dx\ xdx) \xdx) '
Operate on both sides of this equation with —p ,
<f /fcV I. (P fd<o\ h_d^(dm\
df \xdx) x ' da? ' \xdx) x dx \xdxJ
d?_ /dm\ b + 2 d_ / dco\
dx2 \xdxj x dx \xdxj
On comparing this equation with (1) we perceive that we have
here —7- instead of a>, and 6 + 2 instead of b. These changes
xdx
are simultaneous, and if repeated n times the following would
necessarily be the result,
!?(JLY ^!L(A.\U b + 2n d / d\m
df \xdx) dx2 \xdx) x dx \xdx)
60 EQUATIONS NEARLY RELATED
But a = 2n + b,
f ••• -tar- *
81. When the integral of the standard equation is known
for any positive value of a, the integral for an equal negative
value can be deduced from it.
For the equation
d?u _ d?u adu
df dx2 x dx
is immediately reducible to the following form,
d?u _ d „du
(I*
d
dt* dx ' dx
Now operate on this with (xa -j-j ,
df \ dx) ~~ dx' dx\ dx)
If now we write eo for xa y we have
ax
d?(o . d „ d(o
.eft8 da? ' da?
da?8 a; dx
a).
which agrees with the equation in u, with the exception of
having — a instead of a. If therefore u the integral of the
standard equation be* known the integral of (1) will be known
from the equation,
~*s »
It will therefore be a sufficient solution of the problem of
integrating the class of differential equations of the type
d2u _ d*u adu
3? " 5? + 5 d» '
if in subsequent articles we confine ourselves to positive values
of a.
TO LAPLACE'S EQUATION. 61
82. The forms of the following differential equations are
all deducible from the general form
d?u _ d2u a du
dt2 dx2 x dx '
d2u f d . \du
du fa \(m
XW\Xdx+a)dx
^^VdxJ^'dx1
*» = *-** *p (1).
dt2 dx dx '
In this write f for (a — l)t and rj for a?1-*.
*S-A£ »
'• df ' drf
This form fails when a = 1, but in that case the following is
the reduced form ;
2 d2u f d \* .„v
If in this we write f for log x, it takes the following form,
d2u _<* d2u
e
g
dt2 dp
83. The integral of the equation
d2u _ d2u a du bu
dt2 ~ dx2 x dx x2
can also be deduced from that of the standard equation.
Multiply it by x2,
d*u
dt2
(4).
x2
(xd \ fxd , \
m, n being the roots of the equation
m2 + (1 - a) m + b = 0.
62 EQUATIONS NEARLY RELATED
d?u _» /acts m _„ (xd>
x*df ~x
©-•-©*"*
" df "X dx'X ~dx~ {l)'
which it will be observed corresponds to the form (1) of the
preceding Article, m — n + 1 taking the place of a, and uxn of u.
This reduction fails, however, when m and n are equal. In
this case m = J (a — 1),
a*
fxd \*
*S»-©to *
which agrees with form (3) of the preceding Article.
84. To integrate
d*u _ d?u adu
df dx2 x dx
when a is a positive even integer.
In this case on referring to Art. 80 we find that b = 0, and
In = a. Hence the auxiliary equation (1) of that article takes
the following form
d*(o _d2eo
df ~"M*
.-. co = F(t + x) +f(t-x)=F (t +jw).
... USS(A)\
\xdxJ
-&F(t+&
TO LAPLACE'S EQUATION. 63
85. To integrate the same equation when a = 1, i.e. when
the proposed equation is
d2u _ d?u 1 du
dtf do? x dx ' "
2d2u _
n df
^ = (xTx)u'
Now only t can take a minor germ ; but this equation is homo-
geneous on the supposition that t and x are of equal dimensions.
Hence v = - and V, which is a function of v, are of zero dimen-
x
sions. We may therefore assume P = afV to represent any one
of the members of the family of subintegrals. This being
substituted in the proposed equation gives the following equa-
tion for the determination of V in finite terms ;
d?V d
dv2
It is evident that this equation will be integrable if
-(2p + l)«p»,
.-. 0 = (p + l)».
We have therefore to deal with a case of equal roots. One
integration gives the following result,
dV *dV . 17, I>
dv dv
We have now to introduce suppositions, since the form of the
integral of this equation will turn upon the relation between t
and x.
1. If f is less than x2, v* is less than unity, then the equa-
tion to be integrated is
v_ A Bsin^v
64 EQUATIONS NEARLY RELATED
Hence the first subintegral, corresponding to the two sub-
general integrals, is
^sin"1-
(1).
But this subintegral can be raised to zero dimensions by-
integrating it with respect to t, for (a? — f)~^ — -j .sin"1 - (see
Art. 25). We may therefore take the following as the first
subintegral form of zero dimensions,
P = A (sin"1 £) + B (sin"1 £} + B (log mx)\
in which m is an extemporized major germ.
2. Again, let us now take the case when f is greater
than a;2, and therefore v* greater than unity.
The equation to be integrated is in this case,
.-. Vi>2-1. 7=^1 + .B log (vWl + Vf-1),
.-. P = («'-^)-J^+51og(^+l+V/^-l)}...(2).
This integral can be raised to zero dimensions by in-
tegrating it with respect to t, for
As only the variable t takes a minor germ, the family of
subintegrals can be obtained from (1) and (2) by differentiating
or integrating with respect to t.
The following is therefore the general integral required
in this Article,
. -,t
j - sin -
«-J,(l)^-^l+/(l)-TO-if-'><?'
or .-r$)V-W*ffc)*{^l+^l).
ift2>x\
TO LAPLACE'S EQUATION. 65
86. To integrate ^ = J~i + ~a^s* when a u a P°sltlve
odd integer.
Referring to Art. 80 we find that b = 1 in this case, and
n m J (a — 1). Hence the auxiliary equation of that Article takes
the following form, /^^^~^TT^\
d?a> _ d?co 1 da r of the */v
"5? " ^~8 + #3£f U N7 V E R S I T rl
which is the form integrated in the preceding Artidil ,.« k .
/ d \*<*-»
ot- rn . ' dPtl cZ2^ adu ,
87. To integrate -^ = -=— a + ~ T~ when a is not an integer.
The differential equation for the determination of Fin this
case is
do
V d ( tdV\ fo , . dV t . \ XT7
and that this may be integrable so as to give V in a finite form
we must have — (2p + a) =p* —p — ap;
.-. (p + l)(p + a) = 0.
In this example therefore the two values of p are not equal ;
and one part of each subintegral will correspond to p = — 1,
and the other to p = — a.
1. Let^ = -1.
Integrating the above on this supposition we find,
dV 2dV /0 , - n
a
Multiply this by (1 - v*)~l and integrate ;
66 EQUATIONS NEARLY RELATED
We reject the last term, being not integrable in finite terms,
and we only require one integral form for this value of p ;
.-. P = x-1V=Ax1-«(a?-?)%~1 (1).
2. Let p — — a.
Integrating the above equation on this supposition we find
(1 - ff V= A'f(l- v2) dv + B.
We reject the former term of this because it does not give an
integral in finite terms and we require only one integral form
for this value of p ; '
/. P = x'« V=B(x>-?)~% (2).
Gathering the two parts of P together we have the following
complete value of the first subintegral,
P = Ax1-«{x*-ffi~1 + B(<*rf)~*.
The other members of the subintegral family are to be
derived from this equation by differentiation with -g .
If t be greater than a?9, we may write (t* — a?2) for (x2 — f) in
this subintegral.
It will be noticed also that the above subintegrals contain
only even powers of t. Subintegrals containing only odd powers
of t will be obtained from the above by differentiating once
with -j- . We may however pass from P to the general inte-
gral which (as only t can take a minor germ) will be (Art. 19)
88. To find an integral of the equation
-772 + ( cos x - r~ ) V + n \n + 1) cos * • w = 0.
TO LAPLACE'S EQUATION. 67
Only t takes a minor germ; and therefore, changing the de-
pendent variable, we may assume u = emtX, in which m is a
disposable constant
.-. (costf.^)2 X + {n(n + l)costx + m*}X = 0.
An equation of one variable only, of which a particular integral
may be found by assuming X~cos*#, I being a disposable
constant.
.*. Z"sin*a? — lcos*x + n (n + 1) cos8 <c+m2 = Q;
.'. (P+m*) + (n* + n-rt-l)cosix = 0;
.\ Z2 + m2 = 0, and n* + n = l* + l;
.'. l = n or — (n+1), and m = il = in or — i(n + l);
.*. u = Aeint cosw a? + Be"1 (w+1> ' secw+1 a?.
We have introduced this example here chiefly for the
following reason, and it will be hereafter referred to.
The integral of the equation does not depend directly upon
the given value of n, but upon the value of the product n (n+1).
Now this product will remain unchanged if we write — (n + 1)
for n ; and consequently the two terms of the above integral
belong to it of necessity.
The four following Articles are not specially connected with
Laplace's Equation, and therefore do not properly form a part
of the present Chapter ; but are here introduced as illustrations
of the principles laid down in Art. 33 respecting quasi -minor
germs.
89. Equations are occasionally met with which are of the
following type,
*(ax + by, £, |)«-0.
We may simplify this form by writing x, y for ax, by. This
change of the independent variables will reduce this equation
to a form which we may represent by
•("+*as« i)u=0 (1)-
5—2
68 EQUATIONS NEARLY RELATED
and this is the equation which we shall now shew how to
reduce to a more convenient form for integration; our object
being to obtain an equation in which one of the independent
variables shall appear only as a differential symbol of operation
(see Art. 18).
In the equation as now before us, though neither of the
independent variables can take a minor germ, they can take a
quasi-minor germ g ; for the equation (1) is in no way affected
when x + g is written for x and y — g for y simultaneously.
Hence the general integral of (1) must be of the following
form,
u = F(x+g,y-g)
•?*(*r$*M <2>-
And F(x, y) being the first subintegral, all the other sub-
integrals are deducible from it by successive differentiations
\dx dy) '
Now the relation between f^- — -j-j and {x + y) is such
(7 7 x
-z -j- ] (x + y) m 0, and therefore in reference to the
compound operation [-3 -r- 1 the quantity (x + y) is constant.
This suggests the following assumption of new independent
variables f, tj.
Let ^ — cix — hy, and tj = x + y ;
t ,N d d d
and therefore f and 97 are independent variables which satisfy
the above conditions.
TO LAPLACE'S EQUATION. 69
A1 d d • d j d d i d
Also j- = -j- + a -jr. , and -j- = -, o-jt.,
dx drj d% dy drj af
and the equation becomes
w{v'i+ai> i,-bi)u=0 (3)-
from which we see that equation (1) is now reduced to a form
in which one of its independent variables (f ) occurs only as
a differential symbol of operation, and will consequently take a
minor germ.
The following example is one of historical interest.
nA rp . , cPu d?u , 4ta du
90. To integrate ^ = ^s + —s.
We assume f = t — oc, and rj = t + x ;
dV _ a (du du\ n v
•'• d£dv ~v \dv~~dl) '"*' {)'
We may write mf, mrj for f, 9; in this equation without
affecting it ; hence f, rj take a major germ ; and f takes also
a minor germ.
This equation (1) has already been integrated in Art. 70 ;
.\ u = A(jm,7}-a)-a6Jmt (2),
m A (mrj - a)-aem^ + B {mrj + a)-««~**.
Also assuming rf V for the first subintegral, where V is a
t
function of v, and v=-, we find p = - 1, and the first sub-
integral
P = Vrfx = >--a- e^ (4 + i? /is"" V"fl^) (3),
This integral will be in a finite form only when a is a nega-
tive integer. When a is not negative the last term must be
rejected.
70 EQUATIONS NEARLY RELATED
91. An inspection of the integral just found shews that
the case of a = 1 is peculiar, for then it takes the following
form,
w^+s/«-.*);
omitting the last term as not being a finite form, we have
Let this be differentiated with f^J, or (more generally
still) with t(j£;
t-x
= <l>(t + x).et+x.
92. To integrate the equation
, ^d2u , N / du , du\
We here assume x = rj + f , and y—,r\ — \\
dhi _ d2u a — bdua + bducu /1 »
•'• d![2~~dtf+~~ % ~T~^ + ? (;*
If a = b this form of equation has been dealt with in Art.
83 ; but if a and b are unequal we may proceed in the fol-
lowing manner.
The variable f takes a minor germ ; and also we may write
m£, 7nr} for £ 7j without affecting it.
We may therefore assume v — ~ and V— a function of v ;
and the general type of subintegrals is P = rfV; and then
the following will be found to be the differential equation for
the determination of V:
+ 0>9+ op + bp -p -f c) V,
TO LAPLACE'S EQUATION. 71
which will be integrable in finite terms if the coefficients of the
last two terms are equal. This gives the following equation for
the determination of the two values of p corresponding to the
two subgeneral integrals :
p* + (l + a + b)p + (l+a + b + c) = 0.
Represent the roots of this equation by m+jn; their
sum = 2ra ;
/. 2ra = -(l4-a + &);
also the equation in V being integrated once gives the fol-
lowing :
We omit B as not leading to a finite integral, and also
because each value of p, i. e. each sign of jn is required to
furnish only a single integral form of V.
Omitting B and integrating, we find
KV ^ +Br,"(V'-?T} (2).
Also — J(gp.
P may be expressed in terras of x and y as follows :
P.(f-\,+,r{^)\B(^f\ «
and in this case
\dx dyj
93. The preceding Article fails if the roots be equal, in
which case n = 0, and m = — £ (1 + a + b).
72 EQUATIONS NEAELT RELATED TO LAPLACE'S EQUATION.
Let Q = log — — ; we reduce equation (3) as follows; (=
means equivalence) ;
A(J3L)"+B(-2LY-A<« + Bt-
v» + y/ \» + y)
= A (en« + r*) + - (e*Q - €""«)
= A + BQ, when w = 0.
For equal roots therefore
(« + y)- 2 ^ + 5iog-SL) (4).
There still remains the case of imaginary roots, which will
be represented by writing in for jn;
= {AcosnQ + B smnQ] (5).
CHAPTER VII.
EQUATIONS OF THREE INDEPENDENT VARIABLES.
Coefficients constant.
94. All the independent variables of equations of this
class take minor germs ; and therefore a general integral of
any such equation will be expressible in an infinite series,
every term of which contains only positive integer powers
of the variables.
As a general rule the more independent variables are con-
tained in a proposed linear differential equation the more
independent major germs may there possibly be; but this
is not necessarily the case always. A major germ may per-
chance belong to only one, or to two only, or to all the inde-
pendent variables ; and thus an individual independent variable
may be under the influence of so many as there are different
major germs. When the major germs have been introduced
into the general exponential integral (Art. 34), we can then
eliminate them one by one in any order that shall be found
most convenient.
It will be found that the final result of the elimination of all
the germs will sometimes depend upon the manner in which
major germs were introduced into the original exponential
integral ; for sometimes they may be introduced in more ways
than one. And thus we may obtain in more forms than one a
general integral of the proposed equation free from major germs.
95. To integrate the class of equations represented by
d\ _du
dy) dt '
(I
74 EQUATIONS OF THREE INDEPENDENT VARIABLES.
In this class of equations oc, y, t take independent minor
germs, and therefore the general integral u is completely ex-
pressible in a series containing only positive integer powers of
these variables. We may, therefore, assume
tt = P+Q| + ij|1 + -sJ+...
in which P, Q, R, ... are series containing only positive integer
powers of x and y, the general type of them all being the follow-
ing,
Let the above series for u be substituted in the proposed
equation ;
ft .far .***,?** \ p
in which -cr is used, for brevity, to represent <xr ( -y- , -7- J .
Now in the series which P represents the coefficients are,
every one of them, absolutely independent and arbitrary. They
are therefore the coefficients of the family of subintegrals of
which u is constituted. We may therefore replace them with
two independent general germs M, N in the usual manner.
.-. u-(l + * + ?£+...).A*»n
==.^€Mz+Ny+SUV,N)t (1),
which is the usual form of the general exponential integral.
That form of the general integral is therefore proved to hold
good for all equations of three independent variables of the class
proposed in this Article.
COEFFICIENTS CONSTANT. 75
C7 7 v
-T-, -j- J not resolvable into equal
factorials, for such a case requires a somewhat different treat-
ment. See Art. 69.
96. To integrate the class of equations represented by
(d d\ _anu
*W dy)U~df'
the operative symbol st f-r- , -t-j being supposed to be not re-
solvable into any factors that are equal.
Following the method of the preceding Article we find in
this case,
P representing the same series as before, and Q being an in-
dependent series of precisely the same form.
The two terms of which u consists are the subgeneral inte-
grals, one of them containing only even, and the other only odd
powers of t ; and wo notice that the form of the latter subgene-
ral integral may be deduced from the former by differentiating
with -=- once.
at
Now for the same reason as in the preceding Article we
may assume, as was there proved, that
Let *T{MyN) = L\
/ft* \
. • . first subgeneral integral = [1 + ^-is7+t^'dt2 + ...J eMx+ir?
-^**(i+£u,+Jim-.v")
.*. form of second subgeneral integral is
76 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Both of these terms are comprehended in the one form
j£€Mx+Nyt€jLtm
Hence both the subgenera! integrals may be included in the
single form
u = Ae*******1*, subject to L2 = w (if, N).
Hence the exponential integral of Art. (30) is general and
complete for all linear differential equations that belong to the
class proposed in this Article.
The existence of the two independent subgeneral integrals
in this one expression for u is secured and indicated by the sym-
bol j, which always carries with it the double sign ± .
nt_ m . . d*u du du
97. To integrate W = ^ + Ty.
Let the independent variables xt y be changed, the new
variables f , rj being such that x = f + tj, and y — f + mrj.
du du _du , d*u __du .
'*' d^o + dTy~Ti 'and WJi W'
By this change of variables the proposed equation has be-
come an equation of only two independent variables t, f ; and
therefore the remaining variable 17 = y is to take the places
of the arbitrary constants in the integration of (1).
The integral of (1) we find in Art. 60 to be
/. u = AeM^+m=(f)(x-y).eM2^+Mt (2).
This is the general exponential integral; and M is a general
germ.
By the same Article we have the following form of the first
subintegral,
P = A%-ie 4g = y 9)-^H^=g (3).
Jmx — y
The value of P contains only even powers of t, and therefore
it gives us only one of the subgeneral integrals. But the other
COEFFICIENTS CONSTANT. 77
subgenera! integral will be obtained from this by differentiating
with -r (Art. 96). Hence both subgeneral integrals are con-
tained in the following,
\W *Jmx — y
In this m is arbitrary and may be put equal to ( — 1).
98. To integrate £-(£ + £)'«.
Proceeding exactly as in the preceding Article we find
du __d?u
di~~d£2'
... u = <l>(x-y).€m+W (1),
and P=Ji-*e a=y x n • . e 4(m-i)2<»
.: u = F(j>).t-U-Tt.4>{x-y) (2).
93. To integrate g=(J:+-|)V
This may be resolved into the two following independent
simple equations,
du _ (du du\
'di~±\dx+fy)'
which can be integrated in the usual manner.
If we proceed with the proposed equation after the method
of the two preceding Articles we find -gy = -^ , which has been
integrated in Art. 72, whence we shall obtain the integral in its
various forms; but arbitrary functions of x — y will have to be
written instead of the arbitrary constants contained in them.
78 EQUATIONS OF THREE INDEPENDENT VARIABLES.
100. To integrate $ + g + |)\ = 0.
By the same method the integral of this equation will be
obtained from Art. 75.
101. To integrate -55 = -7—7- .
cut doc cLy
The general exponential integral of this equation takes the
following form,
u = Ae*x+Nv+MNt (1)
= AeMx.eN{y+Mi>
= €Mx<f>(y + Mt) (2).
Similarly u = eNv ty (x + Nt) (3),
the last two being the results of the elimination of one germ
only.
To eliminate both the germs from (1) the simplest method
will be to change the forms of the germs M, N by assuming
M = ra + n, and JV= m — n ; m, n being two independent germs.
.*. u = Aem{'x+y)+w'*t . en(z-y)+n2(-t)m
Both m and n may be eliminated by Art. 64; and the chief
subintegral is
_(*+2/)2 OS-*)8 W
This form of P is such that we can at once obtain from it
the form of the general integral u.
-iMiweF *
102. The integral of the equation
du d2u , ,. d2u , d2u
_=_+(a+6)__+a6_
COEFFICIENTS CONSTANT. 79
may be deduced from the preceding Article by assuming as in
Art. 67,
103. To integrate ^| = ^.
The general exponential integral is
u=AeMx+Ny+jLt (1),
subject to the condition L2 = MN.
We may eliminate the germs, and obtain the first subinte-
gral in the following manner,
u = £eMz t eNy+sjN{jt^m.
By Art. 64 this gives the following subintegral by the
elimination of N, _
MP
.-. P = Ay-K€M(x~(y)
This contains only even powers of t corresponding to one sub-
general integral, and the odd powers which are contained in the
other will be contained in -=- . Both subgeneral integrals are
contained in the following formula,
-'(S-'-^O'-S-
*2/>
But x and y are interchangeable in the proposed equation,
and therefore also in P and u. Hence the complete values of
P and u are the following :
80 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Again, since MN=U, we may assume in this case,
M—Le (cosra + isinra),
and N = Le~x (cos m — i sin m) ;
• ^ = ^[gL{(ca;+c~l#) (cosm+t(«c-cr-ii/)sinm+^}
in which Z, m and c are independent real germs.
Let K= (ex 4- ca;"1 ^) cos m +^, and I=(cx — c-1 3/) sin m.
= 4>(K + iI) + ^(K-iI) (3).
104. In the first part of the preceding Article we took no
account of the fact that the proposed equation allows us to
write ex for x, and c"1 y for y quite independently of t The
variables x, y have therefore a special relation, and we have
therefore to consider the following form of the general ex-
ponential integral of that equation,
u — AeL^x+c~ly+j()
= <t>(cx + c-1y+t) + yjr (ex + 0-^-1) (4),
in which c is a general germ.
We may eliminate c by Art. 64, in the following manner :
.'. u = AeiLt . ec (Ljc) +c_1 W ;
= ar*. eL<2V^l2!>
= x~l<l>(2\/xy+jt).
As # and y are interchangeable, the general integral may be
presented in the following form,
u=F (&) • ""* * (2 ^~y +jt) +f (4) • y"** (2V^+i')-(5)-
COEFFICIENTS CONSTANT. 81
,,*- m • , d2u d2U
108. To integrate -5+^- = 0.
We have merely to write — x for x, or — y for y in the
results of the two preceding Articles. Or we may write it for t
106. The integral tf ^ = ^ + («+ &)^ + ai ^ may
be deduced from Art. 103 by means of the same change of the
variable x, y as occurs in Art. 67 ;
x = 7 and v = — r >
a-& u a-b
and
We may derive the following form of u from Art. 104 :
+ £ (a - J) i£]
107. Let a = 1 and b — — 1 in the preceding Article ; then
the integrals of the equation
d*u _ d*u d*u
d?~~df* ~dV*
will be of the following forms,
«=*(S-«-'>->*(W)
82 EQUATIONS OF THREE INDEPENDENT VARIABLES.
But t and 77 are interchangeable, and the two terms of this
integral may be represented as one ; the following is therefore
the form in greater detail,
f + ^-f
^•-'(S-«+«-i#^
♦/©■•***♦ W) *
The second form of u will be the following,
+/(J+yJ).«+io-»+Kf-n*+«ii (2)-
108. If in the preceding Article we write if for f, the
integrals of the equation
d*u (Fit d*u _ n
df+d? + «V~
will be
"'0)-<'+*-'<-^) <■>■
and
« -'(^+-<a) • (>»+*S)-l*(^P+?+6) •(*>
It being understood in these results that 2, f, 77 are all inter-
changeable.
109. To integrate ^ =5^+^ .
This equation takes the form
dt dxdy '
a form which is integrated in Art. 101.
COEFFICIENTS CONSTANT. 83
110. To integrate -73 = ^ — r + u.
0 at ax ay
The general exponential integral of this equation is
u = -4c**+*iH-/X*
subject to the conditional equation
L2 = MN+l or L*-l = MK
We may here assume
Jf->e(£+l)i and iVr = c"l(Z-l),
c being a general germ ;
.-. Mx + JV?/ + j£tf = L {ex + c-1y +j7) + ex- c~xy ;
= Ae!°x-<rlv<l>(ca: + c~1y+jt) (1).
The germ c still remains uneliminated ; we shall therefore
now shew how to eliminate both the germs (L and c) contained
in the exponential integral,
u = j±eJLt t ec(L + l)x+c-HL-l)yt
Hence by Art. 65, eliminating c, the following is the corre-
sponding first subintegral,
p_ €JLt f x-h e±2 <J(L*-l)xy .
Let now £ = 2 V#^, then eliminating L by differentiation of
the last equation we find
. d*(P</x)_d*(P^x)
" -~aT~- dp + *rVf*
and by the method of Art. 78 the first subintegral of this
equation is
G— 2
84 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Hence the first subintegral of the proposed equation is
P = x~i(2^a;i/+jt)-ieLy/t2-^y (2),
whence u is known. The algebraic signs j and + in this inte-
gral are independent ; and oo, y are interchangeable.
111. In the preceding Article c is a general germ; but if
we wish to have the integral which is equivalent to (1) in real
germs, we may write
c (cos m + i sin m) for c, and c"1 (cos m - i sin m) for c~\
c and m being now real germs.
__. . d2u d*u
112. To integrate -^.g-^-*.
Here the exponential integral is
subject to the condition
U = MN-\, or L2-i2 = MJ!T.
We now assume
M=c{L + i), and JST = c"1 (L - i),
c being a general germ;
= </>(ar + c"12/+i^).cos {cx-c~xy + I?) (1).
Following the method of Art. 110 we have
.*. P=eJLt. ^-ie*2V(L2+l)iC2/.
.-. P = a;-i(2A/a;y+i<)-ie±V4^-<! (2),
whence w is known. As before .r, y are interchangeable.
COEFFICIENTS CONSTANT. 85
113. In Art. 74 we Lave shewn that the independent
variables of the equation -j4 + ;p = ° may De changed without
affecting the form of the equation itself. We shall now prove
a corresponding property for the more general class of equations
included in the following form,
s?*^-"^ di)u (1)-
Instead of x and y assume two new independent variables
f, f], such that f = Mx+jmy, and 7) = mx+jMy; the dis-
posable constants M, m being subject to the following con-
dition,
lP±m2 = l .(2).
Let the integral of (1) be u = F (x, y} t) ; and let
W = F& v, t).
df ± dif
«('■ i)
+ ^4- = s> it, %t)W.
But ~M = M W +2Mmdfdv + m W
5 '
and W=™*^ + *M™W + M1W>
d*W d?W d2W , d2W / d
-•(' i)
If.
'• cfo2 * dtf &? ~ dif
On comparing the last result with the equation (1) we
see that W is an integral of (1). And as f, rj contain a germ
that is .not contained in F (x, y, t), the integral F (£, rj, t) will
contain that germ, and be at least as general as Ffa y, t).
Hence without diminishing the generality (and with a
chance of increasing it) we may write Mx+jmy and mx +jMy
for x and y in any integral of equation (1).
In equation (2) we may substitute \ (c + c"1) for M, and
l-(c — c"1) for m when the upper sign of m2 is used in (2) ; but
23
86 EQUATIONS OF THREE INDEPENDENT VARIABLES.
when the lower sign is used we must substitute %{c- c~x) for m,
c being a general germ.
The double sign in this Article is regulated by that in equa-
tion (1).
114. It will be observed as a property connecting the
two sets of independent variables used in the preceding
Article, that
We may represent either of these quantities by r2. Hence
in passing from the equation in terms of x, y to the equivalent
equation in f, rj, and expressing the results in terms of r and
another independent variable, the quantity r will occur in the
two resulting forms in the same manner, and be of the same
value in both.
, n „ ^ . , d?u , d2u du
115. To integrate gp + j^-^.
The general exponential integral may be written in the
following form :
u = AeMx+Ny+(-M2+N**
= AeMH +Mx es*t+y'v.
By Art. G4 the germs M, N may be eliminated, and the fol-
lowing is the general form of the subintegrals,
p = fh € u .t~*e it
&+£ (ix+x/Xix-y)
= t16 U =t1 € U
From this we may find u in the following manner,
\dx dy/ J \dx dyJ
COEFFICIENTS CONSTANT. 87
If now we assume x = r sin 0, and y — r cos 0, this integral
takes the following form, since
y + ix = r (cos 6 + i sin 6) = rei9>
=k^G>) (?>•
116. To integrate
c£V cZ2m _ dw
efo?2 dy* ~ dt '
We might deduce this from the preceding Article by merely
writing iy for y ; but we shall integrate this equation in an
independent manner.
Let f, 7) be a new set of independent variables such that
g = %-t-y and i] = x — y ;
cZ2^ _ du
dgdr)~ 4dt '
This agrees in form with the equation integrated in Art. 101 ;
•••-iKM*1
*M .4!
t,
y*-x*
117. To integrate
d?u dhi _ ePw
dx2 + df~d?'
This has been integrated in Art. 107, but the following method
will serve as an illustration of the variety of ways in which
germs may be introduced into the general exponential integral :
88 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Eliminate iV; then the first subintegral is
(<-a?)-4.€ v *-*'
<->-"(!^f-')
The proposed equation shews that in any integral we may write
jt for t We also notice that x and y are interchangeable.
Introducing these properties, we find the following general
integral,
118. The integral of the equation
&u &u d2u _
will be deduced from the preceding Article by writing it for jt\
This is a complete general integral of Laplace's equation.
119. By a different distribution of the major germs from
that in Art. 117 we may obtain in another form a general in-
tegral of the equation
cPu d?u _ d2u
dx~* + dy*~df'
Let M = ^€^+c-i|+i(C-c-i)|+,Y}
Eliminate c by Art. 65 ; then the following is the first of the
subintegrals,
P = €JLt . (x + iy)-* €±LV*^2
,\ u = (x + iy)-*F(jt±JxT~+tf).
COEFFICIENTS CONSTANT. 89
120. If in preceding Article we write iz for jt, we find the
integral of the equation
d2u d^ d?u_ 0
dtf dtf + dz2'
in the following form,
u = (x + »y)-* J* (Jx?+y2± iz),
which is a form of the general integral of Laplace's equation
agreeing with (2) in Art. 108.
121. To extend to three independent variables the pro-
perty proved in Art. 113 for two.
Our equation is now of the following form,
0
/ d\ d2u d2u , d2u
(fe5jttsa2? + Si + 3? (1)-
Let r2 = x2 + y2 + z2; and let £, v, £ be^
variables such that /p^^^or r^^^f
f=«w+«'y + «"4UlTIVERSiTr
K = ca + c'y + c'V. ! rF0Itf^
The nine constants in these expressions are disposable ; and we
are to dispose of them in such a way that when these values of
f, r]y f are written for x, y, z in any integral of equation (1) the
resulting formula will also be an integral of the same equation.
Let u = F (x, y, z, t) be any integral of equation (1), then is
F= F (f, r], J, t) an integral of the equation
L d\ ,r d2u , d2u d2u
dV dV.dVdV fad , bd , cd\ JT
But ^x = aTi + hl^ + CWr\di + Tv + WV'
d2V__/ad bd cd\2v
•'• dx2 \dg + dr, + dy
90 EQUATIONS OF THREE INDEPENDENT VARIABLES.
f-(f+IM)v
Expanding and adding together the right-hand members of
these three equations, and assuming the following six relations
among the nine disposable constants,
1 = a2 + a'2 + a"2 = b2 + b'2 + b"2 = c2 + c'2 + c"2
and 0 = ab + db' + d'b" = ac + dc + d'c" = bc + b'd + b"d\
we have the following general result,
ffV d? V d2V_d2V d2V d?V
dx2 + dy2 + dz% df + drf + d?
= ^ [t, j^j V, by equat. (2).
Hence V, which is equal to F (£, 77, £ t), is an integral of
equation (1) when the above values of f, rj, f are written for
x, y, z in F(x, y, z, t), which is equal to u.
122. It will be observed that the nine disposable constants
have to satisfy only six conditional equations; and moreover
that those six equations are such as prove the following general
relation between x, y, z and f, 77, f,
x2+y2 + z2=£2 + v2+Z2.
Hence r2, which is equal to x2 + y2 + z2, does not change in
value when we pass from x, y, z to the values represented by
t v, e
123. In reference to the equation (1) of Art. 121 we are
aware that x, y, z which it contains are not necessarily the
co-ordinates of a point P in space, nor is there necessarily any
system of co-ordinates to which they are referred ; but for con-
venience in what follows we will suppose them referred to an
arbitrary rectangular system of co-ordinates Ox, Oy, Oz, and we
will set off upon these axes the values of x,y} z; and suppose P
COEFFICIENTS CONSTANT. 91
the point in space of which x, y, z are thus constituted the
rectangular co-ordinates ;
... 0P2 = r2 = x* + y2+z\
Now on these same co-ordinate axes set off a, a, a" as the
co-ordinates of a fixed point A ; and let b, b\ b" be those of B,
and c, c, c" those of G.
Then if we assume 0A = 0B=0C=1, these assumptions
satisfy the first three of the conditions to be satisfied by the
nine disposable constants ; and if these lines OA, OB, OG be
now supposed to be at right angles to each other, the following
equations shew that the constants will then satisfy the re-
maining three conditions also. For then
ab + a'b' + d'b" = cos A OB = cos ^ = 0,
77"
ac + ac + a"c" = cos A OG - cos ^ = 0,
bo + b'd + b"c" = cos BOG = cos | = 0. .
Hence, if OA, OB, OG be each equal to unity and mutually
at right angles, the six conditional equations are all satisfied ;
and OA, OB, OG may be taken as an arbitrary system of rect-
angular co-ordinate axes. We say an arbitrary system, because
of the nine disposable constants on which the positions of OA,
OB, OG depend three are still disposable, and these render the
position of this system so far arbitrary.
Now as x, y, z are the co-ordinates of P, and OP = r,
x — r cos xOP, y = r cos yOP, z = r cos zOP ;
.-. cos^OP = a- + a'^ + a',- = ^;
r r r r
.-. f = r cos A OP.
Similarly
7) — r cos BOP,
and
£=rcos GOP.
92 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Therefore f, rj, f are the co-ordinates of P referred to the
arbitrary system of rectangular co-ordinate axes OA, OB, OC ;
the position of which in reference to the original fixed system
Ox, Oy, Oz depends upon the values arbitrarily assigned to the
remaining three still disposable constants; which we may in
fact describe as three disposable real germs.
When therefore we have before us an equation of the class
comprehended in the equation
('. D
d \ _ d?u cPu d2u
U~dx~2 + dtf + d?
we are at liberty to substitute, in any integral of it, in the
places of x, y, z, the values of f , tj, f in terms of x, y, z given in
Art. 121, and the formula produced by such substitution will
be an integral of the same equation ; and will virtually contain
three real germs which were not contained in the original
form of the integral. And moreover the value of r2 will not
thereby be affected.
r 7 » rt *« d*ii d*u t cPu
Laplace s Equation, t-2 + -j— a + -p = 0.
124. The following is the simplest form of the general
exponential integral,
subject to the condition if2 4- A72 m 1.
This integral is equivalent to the following form,
u = AeL<**+JM cos (Lz + B).
If we now write in this for x, y, z the values of £, t), £ found
in Art. 121, we shall have a general form which may be thus
represented,
u = A€L(ax+a'y+a"z) cos L (bx + b'y + b"z + B),
LAPLACE'S EQUATION. 93
(d, a', a", by b\ V are not here the same as in Art. 121, but
at present they are disposable).
Let the cosine be replaced by its exponential equivalent;
• u __ j^6L(a+ib. x+a'+ib' .y+a"+ib' - z)^
That this may be the general integral, the following condi-
tion must be satisfied,
0 = (a + ibf + (a + ibj + (a" + ib")\
which separates itself into the two following independent
conditions,
a2 + a"2+a,/2 = &2 + 6'2+Z/'2,
ab + a'V + aW = 0.
Now the presence of the arbitrary germ L permits us,
without any loss of generality, to assume
a2+ a2 + a'/2 = &2 + 6'2+ b"2 = 1 :
and this assumption being made we find, on reference to Art.
121, that these six disposable constants are identical with the
corresponding six in that Article ;
.*. ax + ay -f a!'z = f ,
and bx + b'y + b"z — rj,
and consequently the general exponential integral may be
expressed in the following brief form,
u = AeL* cos (Lri+B) (1),
and the above equations of conditions among the six constants
a, a', a"} b, b', b", shew that f and rj are interchangeable ;
.-. ««4VG'm(2$+.ir) (2).
Either of these results may be regarded as a general integral of
Laplace's equation; or by addition they may be combined
in a single integral.
94 EQUATIONS OF THREE INDEPENDENT VARIABLES.
The six disposable constants are subject to only three con-
ditional equations.
On reference to Art. 1 23 we perceive that f , rj are the co-
ordinates of P referred to the two lines OA, OB; while x, y, z
are the co-ordinates of P referred to the three fixed rectangular
axes Ox, Oy, Oz.
Hence f, rj are the projections of OP (i.e. of r) upon OA,
OP respectively ; and as the positions of these two rectangular
axes are dependent on' three arbitrary germs, therefore the
values of £, rj involve those three germs, and consequently
the integral
u = AeLr> cos {L% + B)
is a germ integral.
If we assign particular values to the three germs we obtain
from this a particular integral: and as particular values of
germs may be infinite in number, we may thus obtain an
unlimited number of particular integrals, which we may deno-
minate a new family of subintegrals : and out of them it may
be possible by proper management to construct a general
integral in finite terms in a form suitable to a physical or
geometrical problem which we may have in hand.
125. To integrate -+--2 + _=0.
Assume the following form of the general exponential
integral,
Eliminate M; then the form of the resulting subinte-
gral is,
-mz2
P= em{iy-x)m {{y + xy i € iy+x
= (iy + x) -h™ V X~W+v) ;
laplace's equation. 95
fr+^'Gnis)
_ J. (x + iy
r \x 4- 1?// \a? + 2^/
-ljr(-£A „.■-!*(•+£) (i)
r \x + iyJ[ r \ r2 J K '
. , 7?i being a germ,
p^i+i V2/«
This integral contains only even powers of z ; conse-
quently this is only one of the subgeneral integral forms.
The other subgeneral form will be found from this by differ-
entiating with -T- .
Let us now assume x — r cos 6 cos <f>, y = r cos 6 sin <f>
z = r sin 6 ;
.'. x + iy = r cos 6 (cos <f> + 1 sin <£) = r cos 0 e?^ ;
.'. w
^i/^ + Mm_^/cos^ V
r \ r2 J r \ r ' J
1 r, /cos 0 ,.\ ,_.
"?'("tt.-^J (3)-
This subgeneral integral contains two independent arbi-
trary functions by reason of the double sign of i.
We may obtain another form of the subgeneral integral in
the following manner :
126. To integrate ^ + ^ + £,= 0.
Assume the following form of the general exponential
integral,
96 EQUATIONS OF THREE INDEPENDENT VARIABLES.
the germs M, m being subject to the equation M* + m2 = 1 ;
.*. U = AeiNz. eM(Nx)+m{Ny\
Eliminate M and m ; the following is the corresponding
form of the subintegral :
P—(a± iy)-h*V ^^W'+iz) .
.-. u m (x ± iy)-*F (Jx* + y* + iz) (1)
= A(x± iy) -* (J a? + f + iz)m,
m being here an extemporized germ. Also y and z are in-
terchangeable.
But x±iy = re*** cos 0, and Jx* + y2 + iz = reie ;
.'. u = A (re*** cos 0)"* {rei9)m
= (re*** cos 6)-* Fire*) (2).
Also u = Ar™-* J cos 6 . €*** . e*1* (3).
The form of this integral differs from that found in the
preceding Article, and illustrates the power we have over the
form of the general integral, since we can introduce the germs
into the general exponential integral in more than one set of
different relationships to the independent variables.
Laplace s Functions.
127. Let the independent variables in Laplace's equation
be changed from x, y, z to r, 0, <j> ; these being defined by the
equations before given, viz.,
x — r cos 6 cos <£, y =r cos 6 sin <f>, z — r sin 6 ;
... ^+^ + / = r3.
and the transformed equation is
9d?u ft du , d*u * du iad7u
LAPLACE'S FUNCTIONS. 97
or more conveniently in the following form,
Concerning this form of Laplace's equation we remark that
r alone takes a major germ ; and <f> alone takes a minor germ.
Also /#,/(/> may at any time be independently written for 8, <j>
in any integral of it.
Now as mr may be written for r (m being a major germ) in
the integral of (1), that integral (Art. 13) may be expanded in a
series in powers of m ;
.*. u = mp .rpWp-\-mq .rqWq-\- ...
and the powers of m in this equation may be replaced with
independent arbitrary constants, which are in fact the coeffi-
cients of the subintegrals of u. The quantities denoted by W
are functions, not of r, but of 6 and cj>.
We may therefore take the following as the general repre-
sentative of the subintegrals of uf
P = rnW.
To determine W corresponding to a given value of n we
substitute this subintegral in equation (1), and the result is,
Q = n(n + 1) cos26 . W + Uos 0 . ~) W+ ~i =0...(2).
This is the Equation of Laplace's Functions, and from this
equation we perceive that r has been divided out, leaving W,
the representative of Laplace's Functions, dependent on the
value of the product n (n + 1), the only remanet of r.
128. Lemma. The numerical value of the product (n -f a)
(n + b) will suffer no change if we write in it - (n + a -f b)
for n.
Hence n (n + 1) remains unchanged in value when - (n + 1)
is written for n.
98 EQUATIONS OF THEEE INDEPENDENT VARIABLES.
This shews that instead of assuming the general form of the
subintegrals in the preceding Article to, be P = rnW, this will
not be a general assumption unless we write Arn -V ar~n~x for rn ;
.\ P=(Arn + ar-n-1)W
is the general form of the subintegrals ; and if it be substituted
for u in equation (1) of the preceding Article the result will
be found to agree with equation (2).
Consequently W, the nth Laplace's function, is also the
-(n + l)th function.
Hence u admits of expansion in two independent series, one
comprising only positive powers of r, and the other only nega-
tive powers ; and the corresponding terms of these two series
will have the same forms of W. We say forms because IF will
contain at least two independent terms, the equation (2) being
of the second order.
129. In Art. 125 we have found the following subgeneral
integral of the equation
° = C0*e-dr{dr+ *J U+{COs6'd6) U + d<t>> '
in which m is a germ.
As a particular case take m—n\ then the general term in
the expansion of u will be
«-£. (cos e.&f.
But we have shewn that this term is equal to Ar^^W;
consequently one part of W is (cos 0 . e^)n ; and the other part
will be found from this by writing — (n + 1) for n ;
.-. W=B (cos d.e^y + b (cos tf.e^r1-1 (1) ;
V. P - {Arn + ar-"-') {B (cos 6 . e^)n + b (cos 6 . e^)"""1},
and if we write m (a germ) for n, we may write u for P.
LAPLACE'S FUNCTIONS. 99
.\ u = (Arm + af™-1) [B (cos 6 . e^)m + b (cos 6 . e^)^1}
= F(r cosO .e^ + ^f (r-lcos6 .?*) (2).
130. We may find Win another general form by means of
Art. 126. For according to that Article we have
u = A (reM* cos 0)~* (reie)m.
Assume m = n + \ as a particular case ; the corresponding
term in the expansion of u will be
rn. (e±i*cosd)-lei(n+W.
Hence the part of W corresponding to this is
W={e±i*cos6)-l€i(n+»9.
If in this we write — (n + 1) for n, we find the remaining part
of IF to be
W = (c*** cos ey* e-^+w.
Hence the complete value of W is
W= (e*** cos 0)~i [Be**+M + be-*n+*»}
= (e*** cos 0)-* {B cos n + £0 + J sin n + |0} (1),
and the value of the general term in the expansion of u cor-
responding to this will be
= (Arn + ar~n-1)W (2).
Thus we have in this and the preceding Article found two
distinct forms of Laplace's functions.
We may write m (a germ) for n, and the results will then
take the form of arbitrary functions, and (2) will be equal to u.
100 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Exceptional case of Laplace' 8 Functions.
131. This occurs when the value of n is such as renders
n (n + 1) = 0, for then the first term of equation (1) of Art. 127
vanishes, and the equation for the determination of W takes the
following form, .
Assume r such a function of 6 as shall satisfy the equation
dO = cos 6 .dr;
.: W=F(T + i<j>)
= *-(e*tan !+^) (1).
Also Arn + ar~{n+l) becomes A + - when n = 0 :
r
,.^ + g^tan^l) (2).
This gives us that portion of u which corresponds to two
terms of its expansion in integer powers of r, viz., A and - .
The algebraic signs of j and i are independent in these
results.
132. To integrate the equation
d\i d2a d2u _
dw* + df+'d?-U'
COEFFICIENTS CONSTANT. 101
We may assume the following as the form of the general
exponential integral,
^ — j^eNMx+NmyJrnz
subject to the two conditional equations
if2 + m2 = l, andiy2 + 7i2 = l;
We may eliminate M and m from this integral, thereby
obtaining the following general subintegral form,
P=enz (x + iy) ~* e^vVts+iVKs-ty)
m (# + iy) ~ i enz+jNs/&+tf,
By the same method we now eliminate n and N, and obtain
the following as the first subintegral,
P= (x + iy) -* . {Jx2 + y2 ± iz) -* &+&++
= (a? + ty)~* (Jx2 + y2 ± w)-» e* (1),
= (r cos 0 . €<*) "* (re**) -I e'r
= r"1 e** (cos 0.el* •***)"* ; (2).
Written out more fully this is equivalent to the following,
r
^(Aer + ae'^j3GC0(Bcos^^ + b8m1^) (3).
It is not to be forgotten that the form of the proposed equa-
tion shews that in integral (1) y and z are interchangeable.
In (3) there are no interchangeable variables.
133. To integrate — +~2+ — = u.
We assume the following as a form which is equivalent to
the general exponential integral,
u^Ae^^+^y+^cosXQx + my + nz + B) (1),
subject to the following condition (which the two disposable
constants p, \ enable us to assume)
(L + il)2 + (M + im)2 +(N + in)2 = 0.
102 EQUATIONS OF THREE INDEPENDENT VARIABLES.
This resolves itself into the two following independent
equations,
L2+M2 + N2 = l2 + m2+n2\
and Ll + Mm+Nn = 0 j ^*
If the above value of u be substituted in the proposed equa-
tion we find that the following relation must hold good between
fju and \,
H2-\2 = l.
Hence if c be an extemporized germ this condition will be
satisfied by assuming
/i = i(c + c_1) andX = J(c-0 (3).
Hence the integral form in (1) is fully determined.
Again, let 8 = Lx + My + Nz, and T^lx + my + nz, subject
to the equations (2) ;
.'. u = Ae^cos\T
= AellS+i?<T
= A 6C • * (S+m +a-1.h(/3- i'D.
Eliminating c from this integral we find the following form
of the general subintegral,
P=(S + iT)-iej^&+T~3 (4).
134. The integrals of the equation
d2u <Fu d2u _
dx2 dy2 dz2
may be deduced from the above integrals by changing the alge-
braic sign of c"1 but not that of c.
-.ok m • , d2u d2u d2u du
135. To integrate ^ + ^+3?-5.
The general exponential integral may be put in the follow-
ing form,
u m AeLz+My+Nz+<-L2+ip+N*)t
L, M, N being independent germs ;
.'. u = AeLH+Lx . €,/2'+if# . eNH+Nz.
COEFFICIENTS CONSTANT. 103
The germs being all eliminated by Art. 64, we obtain the
following form of the first subintegral,
= £-$€~«.
We may pass from this to the general integral in the follow-
ing manner :
A 3 _?.3 -fd . d\ _t±£ \ * -£./& . d\ -*±
3 -I2
= $-*€ «
136. To integrate ^ + ^4.^*^.
The general exponential integral is
11 = AGL (MXz+mN'y+nz+jt)
the germs being subject to the following conditions :
M2 + m2 = l, and N2+n2=l;
.'. U = ^6^^+^) . eM(LNz)+m(LNy\
Eliminating if and m. we find the following subintegral,
P = eL(nz+jt) i ^ + ^-J eLN\/&+y~*
and eliminating iV and ?i in the same manner we find the fol-
lowing subintegral,
P = eL# (a? + ty)"* (J^Tf ± iz)-i^*+++*
= (a? + ty) ~* (Jaf + y* ± iz)-$ eL W+4 ;
104 EQUATIONS OF THREE INDEPENDENT VARIABLES.
/. u = (x + iy)~l (Jx* + y* ± iz) -* F {r +jt) (1)
= r"1 (cos d . €*'<***>) "i F (r +jt)
Vsec 0 / <f> + 0 D . <fc + 0\frT/ N '
• - ^7— (^ cos £|- + 5 sm ^|-J {i^(r + 0 +/(r - t)}.
(2)
137. Laplace's equation is rendered important beyond
most other equations by the circumstance that many pro-
blems of great interest in various branches of Natural Phi-
losophy lead to it, and for their perfect solution render a
knowledge of its general and complete integral a matter of
necessity; for without that knowledge the investigator is
obliged to assume some particular integral known to him, and
this he fixes upon as being likely to answer the object he
has in view. But this is a method which cannot but limit
the generality of his results, and so far limit their authority in
any case of appeal.
We shall, therefore, conclude this Essay with the follow-
ing summary of the method and principles by which we
have been enabled to accomplish the complete integration
of the equation
d?u d*u d2u _
(1). The form of this equation allows all the independent
variables to take independent minor germs.
(2). From this we learn that any general integral of it
may be perfectly expressed in the form of an infinite series in
which the powers of x, y, z are positive integers.
(3). From this it follows that, subject to the condition
the quantities L, M, N being otherwise arbitrary, and not
functions of any of the independent variables, the follow-
ing is a complete and perfect integral of the above equation
of Laplace,
SUMMARY OF METHOD AND RESULTS. 105
When we call this the general integral of Laplace's equa-
tion, we must remember that the word general as thus used
is dependent for its propriety on the fact that Ly M, N are
indeterminate quantities and not mere arbitrary constants.
The equation y2 = x may on the same principle be called
a "parabola" ; but when it is so called we assume that x and y
are indefinite quantities that simultaneously belong to every
individual point of the parabolic curve : for as x, y represent
simultaneously all the values they can possibly have that are
consistent with the equation y2 = x, they simultaneously repre-
sent the co-ordinates of every point of the parabola.
On precisely the same principle we say that L, M, N
simultaneously represent all the values that can be given to
them which are consistent with the equation L2 -f M2 + N2 = 0 ;
and accepting their significance in this general sense we de-
nominate the integral above written the general exponential
integral of Laplace's equation, just as we call y2 = x a pa-
rabola.
(4). There are many systems or sets of values of L, M, N
in terms of two independent germs which will satisfy the equa-
tion L2 + M 2 + N2 = 0. Each set of such values will give a
general integral in a form answering to that particular set
by means of which it is obtained. Examples of this occur in
the preceding Articles.
The following set of values of L, M, N containing two germs
only, leads to a simple result ;
2Lx + 2My + %Nz
= (L - iM) (x + iy) + (L + iM) (x - iy) + 2JV*.-
Let H2 = L-iM, and K2 = L + iM;
.-. H*K2 = L2 + M2 = -N2)
.-. N=iHK.
And 2Lx + 2My + 2Nz =H2{x + iy) + K2(x- iy) + 2iHKz ;
.-. u^AeWW cos {(H*- K2) y+ 2HKz + £)...(«).
E. 8
106 EQUATIONS OF THREE INDEPENDENT VARIABLES.
Also u = Aem(x+iy)+2HKisi+Ki(x-iy).
From this form of u we may first eliminate H and then K,
which will give one part of us first subgeneral integral. The
other part of that same subgeneral will be obtained by first
eliminating K and then H. The result of elimination is the
complete first subgeneral
%y
The second subgeneral may be obtained from this by dif-
ferentiating with -j- . (See Art. 125.)
(5). We are tempted to pronounce the integral just found
to be perfectly general, and so it is in one sense, because it is
the type of the missing terms. But in another sense it is not
the complete general integral, for the form of the differential
equation of which it is the integral shews that in the complete
general integral x, y, z must be interchangeable ; they must
therefore be made to enter the general integral in a symmetri-
cal form.
This we may accomplish by means of Art. 121, and the
result will be that we shall have the following instead of the
integral in (4), (see Art. 125, equat. 1).
ru = F[Ax + BJ + Cz),
A, B, G being germs subject to the equation
A2 + B*+C2=0.
We may therefore write L, M, N instead of A, Bt G.
But in (3) we have the following form of integral in terms
of the same quantities,
= F(Lx + My + Nz).
Hence the complete form of the general integral of Laplace's
equation which involves x, y, z in a manner which allows
SUMMARY OF METHOD AND RESULTS. 107
sc, y, z to be interchanged without affecting its form is the
following :
u = F(LX + My + N,) + lf(^±Ml±l£) (/9).
We may therefore announce this as the complete general
integral of Laplace's equation in terms of #, y, z.
(6). We now make a change of the independent variables
from x, y, z to r, 6, cf>.
By this change the differential equation itself takes the
following form :
2 A rd frd , , \ / A d \2 d?u .
CO9e-dr{crr + 1)U+(COS0- dd)U + d?-9-
Also, since 2 cos </> = e** + e"**, and 2 sin <j> = i (e"**— €**) ;
.-. 2Lx + 2My + 2Nz
= 2r (L cos 0 cos </> + M cos 6 sin 0 + JVsin 0)
= ?• ((£ +tif)«-** + (Z - «L2Ef> e^} cos 0 + 2/-^ sin 0
= r (K2e-^ + IT V*) cos 6 + 2rJV sin 6
- + 2ri HE sin £
If we now eliminate cos 6 and sin 0 by means of their ex-
ponential equivalents, we find
4 (L cos 0 cos <j) + M cos 6 sin <£ + iV sin 0)
r
As iT and K are independent germs, we may omit the factor 4
on the left hand.
Hence assuming Q to represent the right-hand member of
this equation the integral in (ft) may be thus expressed in terms
of r, 0, <j), and two germs H, K>
" = *><?) + */ (|) (7).
108 EQUATIONS OF THREE INDEPENDENT VARIABLES.
By comparing (4) and (6) with each other it will be seen
that the set of germs arbitrarily adopted in (4) is forced on us
in (6) by the assumption, not altogether peculiar to this Essay,
of the following change of variables,
x — r cos 6 cos (f>, y = r cos 6 sin <f>, and z = r sin 6.
(7). If by Art. 24 we express the arbitrary functions in (7)
by means of an extemporized germ m we find the following
equivalent integral,
u = ArmQm + ar""*"^"1.
It is shewn in Art. 129 that the two subgeneral integrals of
the equation
O = m(m + l)cos20. Qm + (cos 6. ^j Qm + ^
(which is the equation of Laplace's functions) are Qn and QTn~l ;
. \ u - (Arm + ar-™-1) (BQm + b Q-^1)
«FW) +]/(|), by Art 24
Hence (A^ + ar^'1) (BQm + bQ-m~1) is exactly equivalent to
the integral in (6) ; and consequently the general expression for
Laplace's mth function is the following,
m* function - BQm + bQ-^1 (S),
B and b being independent arbitrary constants.
The independence of B and b indicates that there are two
distinct Laplace's mth functions, viz. Qm and Q""1"1 ; the product
of which is constant, i.e. independent of m, being equal to Q~l;
where Q = (#2e^ + ZV*) cos 6 + 2iHK sin d
= (He* + Ze'l)2 €*«* + (JBTei: - Ke~^f €*<
(8). There is an exception to (8) corresponding to m = 0,
or — 1. This is pointed out in equat. 1 of Art. 131.
CAMBRIDGE : PRINTED BY C. J. CLAY, M.A., AT THE UNIVERSITY PRESS.
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