92 
PROFESSOR DONKIN ON THE 
form (I.), or for the transformation of known solutions into forms convenient for the 
application of the method of variation of elements — by making the discovery of prin- 
cipal functions depend upon that of n integrals satisfying given conditions, rather 
than upon the solution of a partial differential equation. Having now prepared the 
way for this inquiry, I shall proceed with it in the following section. 
Section II. 
21. Theorem. — If p, q, r be any three functions whatever of the 2 n variables 
vr„ ..., x n , I/,, ...y n , then 
[0> q], r]+llq> />]+[[/, p], q]= o (30.) 
(The symbols have the same signification as in the last section. See art. 9.) 
This may be proved as follows. It is evident that if the above expression were 
developed, each term would consist of a second differential coefficient of one of the 
functions p, q, r, multiplied by a first differential coefficient of each of the other two. 
Consider then the terms in which .p is twice differentiated ; these will be of the 
three forms 
d 2 p dq dr d 2 p dq dr ^ ^ d 2 p dq dr 
dxidyj dxj dip dx t dxj dip dijj dipdyj dx t dxj 
each of which will arise from the first and third terms of (30.) only. (It is to be ob- 
served that i may =/.) 
Now if we examine each of these forms, w T e see easily that for every term arising 
from the first term of (30.), there is a similar term with the opposite sign arising from 
the third term of (30.) ; and since a similar proposition would be true of the terms in 
which q, r, respectively, are twice differentiated, the whole expression on the left of 
the equation (30.) vanishes identically. The theorem is therefore established. 
It is obvious that p, q, r may contain, explicitly, any other quantities (as t) besides 
the 2 n variables with respect to which the differentiations are performed. 
Let | represent, either, one of the 2 n variables x 1} & c., y x , &c., or any other quantity 
whatever, explicitly contained in p and q. It is evident that we shall have 
d 
d£ 
O. ?] = [$>?] + [p 
dq~ 
’ A- 
(31.) 
22. Resuming now the consideration of the 2 n simultaneous differential equations 
discussed in the first section, namely, 
(I.) 
we shall be enabled, by means of the theorems (30.), (31.) of the last article, to give 
a very simple and direct proof of the proposition indirectly demonstrated in art. 9. 
For let u be any function whatever of the variables x x , See., y Xi &c., t; then 
