268 Transactions of the Royal Society of South Africa. 
Exactly similarly it is found that 
¥(a b c d ef g h) = (e—a)(f—b)(g—c)(h—d) . F(6 c d ef g), 
and we thus finally reach 
¥{a b c d ef g h) = (e—a){f—b){g—c)(h—d) 
• (e-b)(f-c)(g-d) 
■ (e-c)(f-d) 
. (e-d). 
4. The marked resemblance of this development to that of the alternant 
| a 0 b 1 c 2 d* e 4 | 
cannot but strike the student, the factors in both cases being all differences, 
and the number of them in the one case being the same as in the other. 
We are thus led to expect that the said alternant is derivable from 
F(a, b, . . ., h) by specialization ; and such is readily found and shown 
to be the case. For, returning to the development at the close of the 
preceding paragraph, and putting 
f,g,h = a, b, c 
we obtain 
(e— a)(a— b)(b— c)(c— d) 
. {e—b)(a—c)(b—d) 
. (e—c)(a—d) 
. (e-d). 
i.e. (-1) 6 . (b-a)(c-b)(c-a)(d-c)(d-b)(d-a)(e--d)(e-c){e-b)(e--a) s 
so that our result is 
¥(abcdeabc) = \a°b 1 c 2 d* e 4 | = f*(a b c d e). 
5. An interesting alternative mode of proving this last equality is to 
show that 
¥(abcdeabc) . | a 0 b 1 c 2 d 3 e 4 | = | a 0 b 1 c 2 d* e 4 | 2 . 
Taking for shortness' sake determinants of the 4 th order we have 
1 
ab-\-ac-\-bc 
abc 
a 3 
—a 2 a 
— 1 
1 
b+c+d 
bc-{-bd-\-cd 
bed 
6 3 
-b 2 b 
-1 
1 
c-\-d-\-a 
cd+ca-^-da 
cda 
c 3 
—c 2 c 
-1 
1 
d-\-a-\-b 
daArdh^ab 
dab 
d* 
-d 2 d 
-1 
{d-a)(d—b)(d—c) 
(a—b)(a—c)(a—d) 
(b—c)(b—d)(b—a) 
(c—d)(c—a)(c—b) 
= - | a 0 b 1 c 2 d s | 2 ; 
