84 
PROFESSOR DONKIN ON THE 
other quantities whatever, except a x , &c. It is assumed that these equations are 
algebraically sufficient to determine each of the n variables y x , ... y n , as a function of 
the other n variables x x , ... x n and the constants. Then the theorem in question is 
as follows : — 
If, by means of the equations (a.), the n variables y x , ... y n be expressed as 
functions of x x , Se c,, then in order that the conditions 
dyi_dyj 
dxj dxi 
may subsist identically, it is necessary and sufficient that the expression [a,-, a,] 
(defined as in art. 9.) shall vanish for every binary combination of the n equations. 
This may be proved as follows : — - 
Putting h, k for any two of the constants a x , a 2 , Sec., let h=cp(x 1 , &c., y x , See.) repre- 
sent one of the equations (a.) above written. If in this equation the values of y x , .-.y n 
be expressed, as above supposed, in terms of x x , Sec., a x , Sec., it becomes identical. 
Differentiating it, on this hypothesis, with respect to x t , we obtain 
dh dh dy x dh dy 2 dh dy n 
dx t ' dy l dxi dy% dxi dy n dx t ' ’ 
and in like manner 
dk dk_ dyj dk_ dy% dk dy n 
dxi dy l dxi dy 2 dx;.'"' dy n dx t ’ 
and if we multiply the first of these equations by ^ and the second by ^ and sub- 
tract, there results an equation which may be written as follows : — 
dh dk 
dy t dx t 
dh dk v | d'!)j / dh dk dh ^ \ 1 
dx t dyi j\dxi\dyj dy t dy, dyj J ’ 
or, putting now a p , a q instead of h, k, and employing the same notation as before, 
d(a,p, a q ) f dijj d(a p , a q ) | 
%i 5 x d ~ j \dx{ d{yj, y { ) J * 
If now the terms on each side be summed with respect to i, the result on the first 
side is [ a p , o ? ] ; and observing that on the second side the term multiplied by ^ will 
only differ in sign from that multiplied by we shall have 
the summation on the right extending to all binary combinations i,j. Suppose this 
equation to be written at length, and then after multiplying each side by 
y s ) } 
d[a p , a q )’ 
let the sum be taken with respect to all the binary combinations p, q. It follows 
