74 
PROFESSOR DONKIN ON THE 
These two theorems (expressed in a different notation) may be found in the memoirs 
above cited. But the following, which we shall have occasion to employ hereafter, 
has not, so far as I am aware, been explicitly stated. 
Inasmuch as = 
dyi 
dyi 
dyj 
=0, it follows that the determinant represented by 
y« 2 ; ••• y am ) ^ 
— y/O 
is = l if (3 15 |3 2 , ... (3 m be the same combination of indices as a ,, a 2 , ... ot m , but is =0 in 
every other case. (For in the first case the determinant is formed with 1 , 0, 0, ... ; 
0, 1, 0, — ; 0, 0, 1, ... ; &c., but if there be one index (3 t which is not contained in 
the series a 15 a 2 , &c., then one row of terms in the determinant will consist wholly of 
zeros.) 
Now considering y l} y 2 , &c. as functions of x l9 x 2 , &c., and again considering these 
latter as functions of y 1} y 2 , &c. given by the inverse equations, we have, by the prece- 
ding theorem, for the value of the determinant (E.) above written, the expression y„= 
^ J ^(y«i? y« & ••• y*m) ••• ^ym) | 
m , x w ••• y& ••• yp m )i 
(where « 15 a 2 , ... a m ; j8 15 ^3 2 , ... j8 m are two determinate sets of m out of the n indices, 
and the summation with respect to the indices y extends to every combination of m 
out of the n). Consequently, 
V m =l or Vm=0, (3.) 
according as the series of indices 
15 ft 2 , ... (3 m 
is, or is not, the same combination as 
a l, a 25 •••• a ;n* 
(I suppose, for convenience, that when the two combinations are the same, the arrange- 
ment is the same in each ; otherwise the value of may be — 1.) 
This is the theorem in question. If we put m= 1, we obtain the equations (1.) and 
(2.) given at the beginning of this article. If we put m=n, the expression y» reduces 
itself to the product of the two determinants formed respectively with the complete 
dy. dx % 
sets of differential coefficients ff &c., &c., the value of which product is =1, as is 
well known. 
As an illustration , it may be useful to exhibit the theorem in the case of m=. 2, 
as expressed by the common notation. Namely, 
_y{ l fyp fyq dy p dy q \ / dxj dxj dj^d_xA'\_ _ 
2 [ \dx dxj dxj dxi J \dy rx dy$ dy$ dy a ) J 5 0I * : 
. . (4.) 
according as a, (3 are, or are not, the same as p, q. Here a, (3 ; p, q are two deter- 
minate pairs of indices, and the summation refers to i , j , extending to every binary 
combination. 
