72 
PROFESSOR DONKIN ON THE 
this equation would become identical if x x , x 2 , ... x n , in its second member, were 
expressed in terms of y x , y 2 , ... y n \ hence, differentiating each side, on this hypothesis, 
first with respect to y h and then with respect to y j} we obtain 
^ _dy^_ dx^^dyj. dx ^_dy^ dx^ ^ 
dx j dyi'dx 2 diji ‘ dx n dyi ' ^ 
d^j_dyi_ dx^j^ j_dyi (c> ^ 
dx x dyj dx^ dyj' ' dx n dy/ •••••••. 
where j is any index different from i. These theorems are given by Jacobi in his 
memoir “De Determinantibus functionalibus.” They are however only particular 
cases of more general theorems, which may be investigated as follows. 
If we represent by 
b j •> •••• 
p, q, r, .... 
any two determinate sets of m indices each, selected out of the series 1, 2, 3, ... n, 
then the determinant formed with the m 2 differential coefficients 
dj/i diji 
dx p dx q 
fm, d Vi, . 
7 * 7 5 • • • 5 • 
ax p ax q 
possesses properties remarkably analogous to those of a simple differential coefficient. 
This analogy was pointed out by Jacobi, and has been further developed by M. Ber- 
trand in his “ Memoire sur le Determinant d’un systeme de Fonctions” (Liouville’s 
Journal, 1851). 
It appears to me that such functional determinants might be appropriately and 
conveniently denoted by a symbol analogous to that of a common differential coeffi- 
cient ; thus 
Vi, Vm .y, 
X n * X r 
d{0Cpy c 
and I shall adopt this notation in the present paper 
For example, 
d[u, v) 
d (x, y) 
would represent the determinant 
du dv du dv 
dx dy dy dx 
[The expression (D.) is not a mere arbitrary symbol, but, like a simple differential 
coefficient, is a real fraction. For if we denote by 
Xgy X f) ...) 
the determinant formed with the m 2 quantities 
d x x p , d^Xqj d \ i ;- 5 i 
