1899-1900.] Dr Muir on Jacobi’s Expansion. 
133 
On Jacobi’s Expansion for the Difference-Product when 
the Number of Elements is Even. By Thomas 
Muir, LL.D. 
(Read March 19, 1900.) 
(1) The character of Jacobi’s expansion of this form of alternant 
will be more readily understood if the two simplest special cases be 
first considered. 
Taking then the case of the 4th order, we have according to 
Jacobi, 
@{abcd) = (b - a)(d - c)(a 2 b 2 + c 2 d 2 ) - (c- a)(d -b)(a 2 c 2 + b 2 d 2 ) 
+ ( d-a)(c-b)(a 2 d 2 + b 2 c 2 ). 
No proof is given, but there can be little doubt that he obtained 
the result by using Laplace’s expansion of a determinant as an 
aggregate of products of complementary minors. Thus 
l a a 2 a 3 
1 b b 2 b 3 II a 
1 c c 2 c 3 1 1 b 
1 d d 2 d 3 
c 2 c 3 
1 
a 
• 
b 2 
b 3 
r + 
1 
a 
b 2 
b 3 
' d 2 d 3 
1 
c | 
d 2 
d 3 
1 
d 
' c 2 
c 3 
+ 
1 
61 
a 2 
a 3 
1 
b 
a 2 
a 3 \ 
1 
c 1 
d 2 
d 3 
1 
d ' 
’ c 2 
c 3 
1 
c 
a 2 
a 2 
4- 
1 
d ' 
' b 2 
b 3 \ 
and therefore by combining the terms in pairs 
@(abcd) = 
1 a 
1 ^ C 
1 b 
1 1 d 
(c 2 d 2 +a 2 b 2 ) - 
1 a 
1 c 
1 h I (£M 2 +a 2 c 2 ). 
1 d \ 
Applying this process to the next case we have 
1 a a 2 a 3 a 4 a 5 
1 b b 2 b 3 5 4 b 5 
1 c c 2 c 3 c 4 c 5 
1 d d 2 d 3 d± d 5 
1 e e 2 e 3 e 4 e 5 
1 fP P P P 
. | c 2 d 3 eP | 
1 a 
i , 
.j&W/ 5 | + 
