501 
1908-9.] Dr Muir on the Theory of Jacobians. 
kind. Since A* , he says, does not involve differential-coefficients with 
0A- 0A- 
respect to x i , it follows that and cann °t involve differential- 
coefficients taken twice with respect to any one variable. Further, second 
differential-coefficients taken with respect to different variables x , L , x k can- 
not occur anywhere save in* 
dAj + 9A /C 
dX r ; dXr. 
d 2 f 
w J rr 
All we have got to show therefore is that the cofactor of ~ 
& ox, ox, 
m 
k 
0A- 0A . . . 
x — 1 + x— * vanishes. To do this we express A, in terms of the elements of 
OXi ox k 1 
one column and their cofactors, say 
A Stl 
A, = a, V-+ + 
dx, 
dxi. 
+ 
+ a, 
¥n 
1 dx , . 
and thus know as above that 
A Vi 3/a 
A k = - 0-1^— - ~ • • 
dx. 
dx, 
a. 
¥n 
' dx , 
where oq , a 2 , • • • involve no differential-coefficients taken with respect to 
x i or with respect to x k . 
The observation made in the course of the first proof that A , A 1: . . . , A n 
are themselves functional determinants leads Jacobi to the conception of 
“ partial functional determinants ” on the analogy of partial differential 
quotients. The fundamental lemma then becomes viewable as the analogue 
of 
0^i 
dy 
dx dy 
dx 
= 0, 
or, in Jacobi’s words, “ gravissimam manifestat analogiam determinantium 
functionalium et quotientium differentialium partialium.” 
Apparently this recalls to Jacobi another analogy of the same kind, 
which he had omitted to draw attention to in his paper of 1830, when the 
first two cases of the lemma had been originally enunciated by him. The 
proposition involving the said analogy he now generalises thus : — If f , f \ , 
f o , . . . , f n be expressible as series the terms of which involve only powers 
of the variables x , x x , x 2 , . . . , x n , the functional determinant does not 
* It would have been well to make clear here that every term of the final expansion of 
2 0 A ■ 
— 1 contains one and only one second differential-coefficient. 
000 % 
