502 
Proceedings of the Royal Society of Edinburgh. [Sess. 
involve a term in x _1 x x 1 x 2 1 . . . . x n \ In support of it he has only to 
point out that the functional determinant is equal to 
S(/A) + 8(/A 1 ) + _ _ _ + 9(/A n ) 
dx dx 1 dx n ’ 
and that the development of the k th term of this expansion cannot contain 
a term in . 
After referring to a possible application of the lemma in connection 
with definite multiple integrals, Jacobi concludes § 2 by returning to the 
lemma itself and throwing it into a third form originally announced in 
1841 ( Be determ, funct. § 9). Viewing x , x x , x 2 , . . . , x n as functions of 
} f , . . . , f n he obtains of course {Be determ, funct. § 8) 
dx A dx. A, dx n A„ 
d/~n’ d/~n’ ‘ ’ W ~ r’ 
so that by substitution the lemma becomes 
or 
0 = 
<fg 
dx 
dx R 
df dx 
jdx 
fa- a 0 *! 
' I: . ,, OJ , df 
+ K [d^ + ^ + 
3/ 
0 
dx 1 
f)x fx l 
d log R df d 0 / 
9/ + aV + a^ 
+ 
a ( R v) 
aA— i 
. V df) 
cx n 
dx ± 
+ . . 
) dx, dR] 
dx n BR) 
+ . . 
df dx Y 
9/ dx n 
> dx, 
d — - 
f 
0 9 Xn v 
+ R ^ 
+ . 
. . + K-i 
, r J 
dx n 
11 y 
idx 
+ 
K) 
9/ ! > 
'71 J 
dx. 
+ 
J 7 
dx,. 
where firstly R and the as’s have to be viewed as functions of the /’s and 
all differentiated with respect to /, and secondly the differential-coefficients 
thus obtained have to be expressed in terms of the cc’s preparatory to 
performing the final set of differentiations. 
Here the consideration of functional determinants would have come to 
an end in the present memoir, had it not been that a theorem on the 
subject which had been given incorrectly in 1841 (Be determ, funct. § 14) 
was now wanted in § 3 for use. Two and a half pages (pp. 215-217) of 
matter are consequently intercalated in order to enunciate the theorem 
