Determinant with Factors like those of the Difference-Product. 269 
and consequently the multiplicand 
= -| a 0 b 1 c 2 d 3 \. 
It is also just worth noting that the one determinant can be directly 
transformed into the other, the first step of the process being to perform 
on F (a b c d a b) the operation 
col 2 — (a-\-b-\-c-\-d) colj. 
6. Besides the specialization 
f,g,h = a, b, c 
there are a number of others resembling it although of less interest : for, 
without causing the function to vanish, we may put 
f = a or e, 
g = a or b or e, 
h = a or b or c or e ; 
for example, 
¥{abcdeeee) = {e-a){e-b) 2 {e-c) 3 {e—d) i . 
7. Of far greater interest, however, is a specialization of a different 
kind, namely, that which leads to a new determinantal representation 
for the product of the binomial sums of a, b, c, . . . It is well known that 
for representing such a product as 
(a-\-b){a J r c){a J r d) 
• (b+c)(b+d) 
. (c+d) 
there is no form analogous to the alternant, and that in fact when the 
said product does occur in determinantal analysis it appears as a quotient, 
namely, 
| a 0 b 2 c 4 d & | -r | a 0 b 1 c 2 d 3 \. 
8. Going back to the result of § 3 and altering the signs of the second 
triad of variables, we obtain 
F(a, b, c, - d, -e, - f) = {-d —a)(—e ~b)(-f -c) 
. (—d -b)(-e -c) 
. (—d -c) 
= (-1) 6 . (d +a )(e+6)(/+c) 
. (d+b)(e+c) 
• (d+c) ; 
and similarly for all other orders of F. In other words, By altering the 
