86 
PROFESSOR DONKIN ON THE 
total differentiation with respect to t) there be given n integrals, involving n arbitrary 
constants a x , ... a n , as 
Pii.^ 1J ^25 ••• 3/l5 ••• 3 /b> Os 
7l( — 1 ) 
the remaining integrals may be found, whenever the — 5 — conditions [a ; , a,-]=0 are 
satisfied. 
For let y x , ?/ 2 , ... y n be expressed, by means of the given integrals, in terms of 
Xj, ... x n , a u ... Q n , t. 
Their values so expressed will satisfy (art. 12.) the conditions 
dxi dxj 
(b.) 
Let (Z) represent the result of substituting in Z these values of y x , ...,y n , so that 
(Z) is a given function of x 1} ... x n , a x , ... a n , t. We shall have 
d[ Z) d7i dZ, dy x d7i dy 2 
dxi dxi)dy x dxi'dy 2 dxi 
which the equations (I.) and ( b .) reduce to 
dxi 
■y>+te l x ‘+s*°+ 
but 
consequently 
/ dyi dyi , dy t , 
d{7i) dyj 
dxi dt 
(c.) 
Looking now at the assemblage of equations ( b.), (c.), we see that they express the 
following proposition : — 
The values of y x , y 2 , ...., y n , -(Z), 
are the partial differential coefficients with respect to x x , x 2 , . 
same function. Let this function be called X ; we have then 
dX_ 
dxi 
x n , t, of one and the 
(II.) 
and since y x , ...,y n , (Z) are given functions of x x , See., a x , See., t, the function X can 
be found by simple integration. 
Let us then suppose X to be known, and let us take the total differential coefficient 
d~X. 
with respect to t, of ; we shall have 
/dX\'_d*X ■ d*X x , d*X \ x , + 
\dcii) daidt'daidxy 1 ' da l dx 2 2 
which, by virtue of (I.) and (II.), becomes 
/ dX\' d(7i) d7i dy. d7i dy 2 
