n

that (1) is reproduced with the same conditions for determining N,N,... N as if π were a symbol of quantity. But the question of its completeness, of its conducting, through the performance of the inverse operations (π — a1)', &c., to the most general solution of (1), is one that we are not called upon to determine a priori. In all the cases we shall have to consider, its completeness will be obvious.

du

dx + (x2 + x − 1) u = 0

Let (1)10=y, then, since (π-1) y = 0, we have

A very interesting application of the same theory to the solution of partial differential equations is afforded by what Mr Carmichael has termed the index symbol of homogeneous functions. Cambridge and Dublin Math. Journal, Vol. VI. p.277.

Since, if u represent a homogeneous function of the ath degree of the variables x,, x,,..., we have

it follows that, if we represent the symbol x1

and therefore, in accordance with the reasoning of Arts. 3 and 4,

an equation of which the second member expresses the complete, because the only, value of the first member when ƒ (π) is rational and integral, but a particular value when the first member contains inverse factors.

n-2

Hence, if we have any equation f (π) u = X, where ƒ (T) is of the form 7"+‚π”¬1 + à ̧π” ̄... +A,, and X is a series of homogeneous functions of the variables, suppose

we get

X = X2+ X2+ ... &c.,

a

u = {ƒ (π)} ̃1X +{ƒ(π)} ̃10

=

{ƒ (π)} ̄1Xa+ {ƒ (π)} ̄1X, ...+ {ƒ (π)} ̄10

= {ƒ (a)} ̃1 X2+ {ƒ (b)} ̄1X, ... + {ƒ (π)} ̄10, by (A).

a

i

To find the value of the last term, we proceed, as in Art. 5, to reduce it to a series of terms of the form A ̧ (π — a)−10, i being the number of roots equal to a of the equation f(m)=0. Now it may, by an induction founded on successive applications of Lagrange's method for the solution of linear partial differential equations of the first order, be shewn that

(π-a)-'0 = u(log x,) + v. (log x,)...+w....(B),

Ua Vawa being arbitrary homogeneous functions of X1, X2, Xn of the ath degree.

To this result we may give the symmetrical form

(π − a) ̄' 0 = u ̧L1 ̈1+v ̧M*→ ...+wa›

L, M, &c. being logarithms of any homogeneous functions which are not of the degree 0.

It remains to shew how it may be ascertained whether a proposed partial differential equation can be reduced to the form f()uX.

d

Let us resolve each symbol

dx'

, entering into π, into two,

d

dx.

and let represent as operating on x, only as entering

dx,

But as ", in (C), operates on the variables only as entering into, which is a homogeneous function of those variables of the first degree, we may replace it by unity. We have therefore u= (-1) πU. In the same way it may be shewn that 'u(Tr+1) (π − r + 2)... πu. And thus it is seen that any partial differential equation which is expressible in d d the form f(T) = X, on the hypothesis that &c. de' dx' operate on the variables only as entering into u, is reducible to the form (7) u = X, independently of such restriction. This reduction having been effected, the solution can be found by means of (A) and (B), whenever the second member consists of one or more homogeneous functions of x1, X2, ... X'n

1

Therefore {π (π − 1) − nπ + n} u = x2 + y2 + x3,

or

whence

u = {(π — n) (π − 1)} ̃1 {x2 + y2 + x3} + {(π − n) (π − 1)}~1 0

=

u, v, denoting arbitrary homogeneous functions of the degree n and 1 respectively.

10. We may, by simple transformations, reduce to the above case various other classes of equations differing from the above only as to the form of ; e. g. the class in which ; but, passing over such special forms, we shall consider the general equation ƒ(#)u=X,

where

d

dx,

d

...

d

π=

d + X2 dx,

2

...

and each of the coefficients X,, X,,... X, as well as X, may be any function whatever of the independent variables. And we design to shew, first, how it may be determined whether a given equation admits of reduction to the more general form above proposed; secondly, how, then, to integrate it.

Suppose the given equation of the nth order; then the symbolical form in question, should the proposed reduction be possible, will be

Now the highest differential coefficients in the given equation will arise solely from the symbol ", and the terms in which they occur will enable us to determine the form of π. Thus, for two variables, we have

were sym

as they would be, if, in the first member,

d d dx' dy

bols of quantity. And this law is general for the highest differential coefficients.

Again, the form of being determined, the values of A1, A,,... will, whenever the proposed reduction is possible, be found by effecting the operations implied in the first member of (4), and comparing with the first member of the equation given.

Suppose the equation reduced to the form (4). Then, if the auxiliary equation

m" + A ̧m"-1 + A ̧m"-2. + An

...

=

......

have its roots all unequal, we have a series of terms of the form (a)X; and each such term involves the solution of a partial differential equation of the first order of the form

du

dx2

But, if the auxiliary equation (5) have equal roots, partial differential equations of higher orders will present themselves. We deem it therefore important to shew how this difficulty may be avoided, or, to speak more precisely, how its solution may be made to flow from that of the corresponding case of linear differential equations with constant coefficients.

...

Introduce a new system of independent variables y1, Y2,Y

d dy,

so conditioned as to give π = To prove that such a sys

Yn' in

tem exists, and to discover it, let us assume y1, Y2,... succession, as subjects of the above symbolical equation, and examine whether the results are consistent. And first, assuming y, as subject, we have