132 Proceedings of Royal Society of Edinburgh. [sess. 
(i a - b)(a - c) . . . . (b - a)(b - c) .... (c - a)(c - b) . . . 
an expression which, as we have already seen in the preceding 
paper of Jacobi’s,* is equal to 
As the left-hand side, &PP', becomes under the same circumstances 
k. P 2 , 
we have as our last desideratum 
and are thus enabled to formulate the proposition 
2[ + (x-a)(y-b)(z-c) . . . .1 
p tey* - : jiv h lWA • • •) j 
' ' (x- a)(x - b)(x - c) . . . (y - a)(y - b)(y -c) ... (z- a)(z - b)(z - c) . . . 
a noteworthy result which in later notation takes the form 
(x-a)- 1 ( x-b)-i- ( x-c )-! L/ iu K (n i Pfaya • • •) • £*(«.&»<=. ■ ■ ■), 
(y-a)- 1 (y-i)- 1 (y-c)-' . . .. \ F(x) . F(ty) . F(z) . . . 
(z - a) ~ 1 (z - b) ~ 1 (z - c) ~ 1 . . . . I 
where n is the number of variables, and 
F(«) = (x - a)(x - b)(x - c). . . . 
* Since Y = F(cc) . F (y) . F(s) . . . . , the first term of the alternating 
aggregate may he written 
g(s) _ g(y) m, 
x- a y — b z-c 
which, on the substitution being made, becomes 
F'(a).F'(&).F'(c) . . . . ; 
and it is this form which in Jacobi is replaced by (-lp^-DP 2 . 
