458 
Proceedings of Royal Society of Edinburgh. [sess. 
V M = 
Ql1.7 
+ ^8,6 
+ C 6,6 
V 8l8 = 
^13,8 
+ ^10, 7 
+ C 8,7 
V 9 , 9 = 
^15,9 
+ ^12.8 
+ C 10,8 
^10.10 = 
: ^17,1C 
> + 0,4.9 
+ C 12,9 
V„, n = 
: ^19.11 
. + ^16.10 
i + ^14.10 . 
From a consideration of these it might possibly occur that the 
general expression for Y n n was 
P i n I p 
2n - 3,n “ r 2n -6,n-l ^2fi-8,w-l' 
The very next case, however, would serve to disprove it, for 
^ 12.12 = ^ 21.12 + ^ 18.11 + ^ 16,11 ” ^ 11 . 11 » 
the expansion of (1 - a?)(l - x 2 ) . , . . (1 - x 12 ) being 
1 - x - x 2 + x b + x 7 - a? 12 + . . . . 
and .*. 
^ 12, 8 ~ ^ 8 + 11, 11 “ ^ 8 + 10, 11 “ ^ 8 + 9, 11 + ^ 8 + 6, 11 + ^ 8 + 4, 11 “^ 8 - 1, 11 + * * * 
and V 12>12 , as stated. The expression given for V n should there- 
fore he reinforced by the term - C„ „ if it is to he correct for 
n= 12. And no alteration in this is necessary, save by way of an 
addendum, if we proceed to make the expression suffice also for 
n = 13, 14, . . . . For, having got the expansion of 
(l-x)(l-x 2 )(l-x B ) (1-a 12 ) 
to be, as far as is needed for this purpose, 
1 - X - X 2 + X 5 + x 1 - X 12 + . . . . 
we see that the multiplication of it by 1 - x 1B will not affect the 
coefficient of x 12 , the expansion of 
(l-x)(l-x 2 )(l-x B ) (1-a 13 ) 
being in fact, as far as is necessary, 
1 - x - x 2 + x b + x 1 — x 12 + O'# 13 + . . . 
from which we have 
"^ 13,8 = ^ 8 + 12,12 “ ^ 8 + 11,12 “ ^ 8 + 10,12 ^ 8 + 7,12 * ^ 8 + 5,12 ^ 8,12 ^ ^8 - 1,12 ‘ 
