500 Proceedings of the Poyal Society of Edinburgh. [Sess. 
row of R along with each element of the second row, thus forming (n + 1) 2 
products whose cofactors in the determinant he denotes by 
(00) (01) (02) . . . (On) 
(10) (11) (12) . . . (In) 
( 20 ) ( 21 ) ( 22 ) . . . ( 2 n) 
(nO) (nl) (?z2) . . . (nn) 
— that is to say, he puts 
(lk) for the cofactor of 
a notation which necessitates 
d in R 
dx { dx k 
5 
(Hi) = - (Id) and (it) = 0 . 
It follows on this that the cofactors of 
dj\ dj\ dj\_ df_ 
dx ’ dXj ’ ’ 0x i _ 1 ’ dx i+1 
in are (i, 0) , (i, 1) , . . . , (i, n) , and 
takes the form 
3ft, 0) | 3(t,i) + . . . 
dx dx-, 
or | {|o)/l , 3{(»,l)/x} + 
dx dx l 
thus the assumption above made 
■ 8 P» = 0 
dx n 
d{(i,n)f A ) = A 
dx n 
Since, however, (i , k)f\ = — (k, %)f x and = 0, we can apply to the 
latter the general proposition that if a ik be any quantities whatever such 
that a ik — — a ki , a H = 0 , and Ef stand for 
then 
dfljo , 90*1 + 
| in 
ox 
dx 1 
dx n 
0H 
0H, 
0H„ 
“f* 
— — 1 + . . . 
+ 
0x 
dXj 
dx n 
as desired. There only then remains to show that the lemma holds in the 
case of two variables, and this is unnecessary because it is then identical 
with the familiar proposition 
= ffi_ m 
dx dy dy dx 
In addition to this gradational proof Jacobi gives one of a different 
00^i* 
dx f - 
dxi 
dx, 
dx k 
* The reason for this, of course, is that 
+ 
= 0. 
