88 
PROFESSOR DONKIN ON THE 
differentiating the equation (X.) with respect to x i5 and employing the equations 
dyp dy q 
dx q dx p 
we have 
d lL\ 4 t I _ 0 
dt dxi dy x dx x dy 2 dx 2 ’ ’ * 
on the other hand, taking the differential coefficient of with respect to t, without 
assuming anything as to the nature of the relations between t and the other variables, 
we find 
- d’Ji , dy { , dyi , 
and adding to this the preceding equation, 
y l+ v=pU-f) +f U-iL ) + ... 
' dxi dx x \ 1 dy x J 1 dx 2 \ 2 dx 2 J 
from which it follows that the n assumptions 
x - = m. 
1 dy t 
would involve the n further equations 
'’ l dxi 
Again, the n assumptions 
dX_ h 
da~ bi 
would give, by combining the n equations obtained by differentiating totally with 
respect to t, viz. 
d*X . d*X 
d*X 
dafC^ dafx^ 1 da t dxf 
with the n others obtained by differentiating the equation (X.) with respect to a„ viz- 
d*X df d*X . df d 2 X 
% 2 ~\~ 0, 
the n following, namely, 
d q X 
da.dx. 
' dy x da i dx l 
' dy 2 da t dx 2 
-r ••• — 
(. df \ 
1 d * X (r’ 
df\ 
V 1 dyj 
' dajdxc, \ 2 
dyj 
from which it follows either that #■= or that the determinant formed with the n 2 
d Vi 
expressions -p- or — (~ ), vanishes; but this last condition would express, as is 
1 daidxj drii \dxjj 1 
well known, the possibility of eliminating the n constants a,, a 2 , ... a n from the n 
equations 
^=^(•*1, & c -> &c., t ), 
which would contradict the assumption that X is a complete solution of the equa- 
tion (X.). 
