508 Proceedings of the Royal Society of Edinburgh. [Sess. 
and the ratio of the determinants of the arrays to be 
y* / + AAk ty_i _ ty' n \ ' 
^ V ~ dx 1 dx. 2 dxj 
To this there is added “ C’est l’expression donnee comme definition par 
M. Jacobi,” — a remark, however, equally applicable when, as at the outset, 
the increments of the afs were taken in their most general form. Pre- 
paratory to the next example it is pointed out that any n of the variables 
x 
2 ’ 
X K 
fi , A 
9 5 
,fn 
may have arbitrary values, the other n being then determinable ; and that 
therefore if n— 1 of them be taken to be invariable, the ratios of the 
increments of the others may be considered known. We thus see that the 
two arrays 
i-H 
rH 
<1 
\x 2 AjflJg . . 
. \x n 
Vi 
0 
0 
0 
0 
^2^2 ^2^3 • ' 
\ 2 x n 
a 2 /, 
^2 / 2 
0 
0 
0 
0 
A 3 ^ 3 
^ 3 /l 
^ 3/2 
^ 3/3 • • 
0 
0 
0 
0 
■ • A n ct n 
^nj 2 
Al /3 • • 
. *nfn 
of the second example are simultaneously possible, the n indepmdent 
variables in the case of the first row being x x , f 2 , / 3 , . . . , f n , in the ( -se of 
the second row x 1 ,x 2 ,f s ,. . . , f n , and so on: and we copsquently^ 'arn 
that the functional determinant may be written in the fori} 1 
(W) (W) . . . ( d A\ 
\0.r 1 / \dx 2 J \d xj 
on the understanding that the brackets enclosing jdx r imply that f r is 
there viewed as a function of x, , x„ , . . . , x r , f r , , , / f . A third 
i ' z 7 7 r ? j 7 -f- 1 * r+‘2 ? * * * 7 J 71 
pair of possible arrays is 
A \ x i 
0 
0 
0 
^l^TO+1 
.... A yl n 
0 
A 2^2 
0 
0 
A^.' m+ i 
^2*^j®+2 ' 
.... A2 x n 
0 
0 
A3X3 . . 
0 
^ 3 X m+\ 
^ 3 ^ 171+3 
.... 
0 
0 
0 
• • ^rrpC'rn 
A„i^ rrt+ i 
^m^'m+2 
.... A m X n 
0 
0 
0 
0 
A m+ i^ m+ i 
0 
. : . . 0 
0 
0 
0 
0 
0 
^m+2^'m+2 
0 
0 
0 
0 
0 
0 
0 
.... A n x n 
