DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
85 
from the theorems of art. 1, that the coefficient of 
dyr_dys 
dx j dx r 
on the right will reduce itself to unity, and that of each of the remaining terms to 
zero ; so that we shall have, writing now j, i for r, s, 
j\y_ d v±— 
dxi dxj 
a 1 . <%» Vi) 1 
d(a p , a q ) J 
(28.) 
In order then that the expression should vanish identically for every binary 
combination of indices, it follows from (28.) that it is sufficient, and from (27.) that it 
is necessary, that each of the — % ^ terms [a p , a q ~\ should vanish, and vice versa. It 
will be observed that the terms [a p , a q ~\ cannot vanish otherwise than identically, 
since they do not contain any of the constants a x , a 2 , & c., and it is by hypothesis im- 
possible to eliminate all these constants from the equations (a.). It follows then that 
when the conditions [a p , aj\ =0 subsist, the values of y x , ... y n expressed as above, are 
identically the partial differential coefficients of a function of x x , ... x n , a x , ... a n . 
We have thus established the theorem enunciated at the beginning of this article. 
13. The preceding theorem may be made somewhat more general as follows: — 
If we divide the 2 n variables into any two sets of n each, so that no two in the 
same set are conjugate (as for instance 
and denote one set by 
and the other by 
X x , X 2 , ... X r , y r + ij J/n 
Hu y 2 ) ••• y r) X r+ 15 ••• X n)> 
±*7l> + %> ••• +*!n 
taking the + or — sign according as ^ represents y t or x t , it is obvious that the ex- 
pression 2; is identical with [ a p , aj\ ; and therefore whenever all the terms 
[ a p , a q ] vanish, if the set tj 13 p? 2 , ••• can be expressed by means of the equations (a.) of 
the last article, in terms of f„ | 2 , ... % n , a x , a 2 , ... a n , their values will be the partial 
differential coefficients with respect to £,, f 2 , ... of a function of these variables and 
of the constants. 
14. Theorem. — If of the system of 2 n simultaneous differential equations of the first 
order 
, dZ ■ _dZ 
X x -j 5 ...., X n 
dy x dy n ^ 
/ dZ I dZ 
y 1 dx x " ’’ dx n - 
(where Z denotes any function of x x , ... x n , y x , ... y n and t, and accents denote as usual 
