186 Proceedings of Royal Society of Edinburgh. [sess. 
qua in formula signo ± substituendum est aut ( - l) n(mrfl) aut 
( - l) m(n+1) prout i par aut impar est.” * 
The second derived identity, — that is to say, the case of the 
final general identity where i = 1, — Jacobi proceeds to utilise for 
the purpose of proving his proposition regarding the effect of the 
vanishing of certain m+1 functional determinants. The path 
which he follows to reach his result is not a little surprising. 
Instead of saying that the vanishing of b, b lt b 2 , ... , b m , — for 
these are the m + 1 determinants in question,— entails the vanishing 
of the left-hand side of the identity, viz., 
•• • p 
(m-i) 
rii ? 
* The fact that these identities can be derived in the way here indicated 
from another which the preceding footnote has shown to be true, not merely 
of functional determinants but of determinants in general, is convincing 
proof that they also ( i.e. } the derived identities) are not restricted to any 
special form of determinant. Using the fundamental identity as enunciated 
in the footnote, and taking the special case of it where n ~ 4 and m = 2 , and 
where therefore the given determinant may be written | a x b 2 c 3 d 4 e 5 f 3 g 7 ( , we 
have 
| a x b 2 c 3 d 4 e 5 | | a x b 2 c 3 d 4 e 6 \ | a x b 2 c 3 d 4 e 7 \ 
I a i h c 3 d 4 f 5 | | a x b 2 c 3 d 4 f 6 | \ a x b 2 c 3 d 4 f 7 | 
! a i c 3 d 4 g$ | I a 4 b 2 c 3 d 4 g 6 | | a x b. 2 c 3 d 4 g 7 | 
\ a 4 b 2 c 3 d 4 \ . | a x b 2 c 3 d 4 e 5 f 3 g 7 | , 
How in each of the determinants forming the first row on the left here, e 
occurs as an element, in the second row f 2 similarly occurs, and in the third 
row g 3 , while on the right these only occur in | a^b^d^^^ | . Consequently, 
equating cofactors of e 4 f 2 g 3 we have 
| a 2 b‘3 C 4 ^5 l I a 2 b$ C 4 d 6 | 
- I a x b 3 c 4 d 5 I - ( a 4 b 3 c 4 d 6 [ 
[ a i b 2 c 4 d 5 | 1 ct 4 b 2 c 4 d 3 | 
which when put in the form 
I a 5 b 2 C 3 d 4 I I a 6 b 2 c 3 d 4 | 
| cl 4 b 5 c 3 d 4 I | & 6 c 3 d 4 | 
| a 4 b 2 c 5 d 4 | | a x b 2 c 6 d 7 \ 
) b 3 c 4 d 7 | 
I a i b- 3 c 4 d 7 | 
I tt x b 2 c 4 d 7 | 
| a 7 b 2 c 3 d 4 1 
| a x b 2 c 3 d 4 1 
| a x b 2 c 7 d 4 | 
2 
| a x b 2 c 3 d 4 | . 1 a 4 b 5 c 3 d 7 | 
| a x b 2 c 3 d 4 | . | a 4 b 5 c 6 d 7 | 
is a case of the first derived theorem. 
The original theorem, it should be noted, is true for all values of n and m ; 
the derived holds only when m<n, — in fact, if we do not, in seeking to 
obtain the latter, take m<n in the former, we shall fail in our aim. Thus, 
taking n = 2 = m in the former, the given determinant being | a x b 2 c 3 d 4 e 5 | , we 
have quite correctly 
\ a x b 2 c 3 \ | a x b 2 c 4 \ | a x b 2 c 5 | 
\ a x b 2 d 3 \ | a x b 2 d 4 | | a x b 2 d 5 \ 
! b 2 e 3 | | a x b 2 e 4 \ | a x b 2 e 5 | 
— I a- x b 2 | . | aib 2 c 3 d 4 e 5 | ; 
