392 Proceedings of Royal Society of Edinhurgli. [june 18 , 
1 a .. . (& + c + (i + . . . + 
1 1) (« + c + cZ + . . 
IIP... (a + & + c + . . .+/.•)"-! 
1 a _ _ Q^n-2^ _ ^Y~l 
I 1) IP .. . {%-hy-^ 
, where %=a + 6 + c + ...+/, 
IIP... 
= 0-0+ . . . 
(5 - ly-^ 
+ 
I a .. . a^-\ (-af~^ 
I b p .. . {-by-^ 
IIP... p-^, {-iy~^ 
= (-iy-\ ^(abc. . . Z); 
which establishes the identities for integral functions of A. 
8. Before proceeding to prove the results involving fractional 
functions, we observe that the identities also hold for another series 
of fractional functions ; namely, in the case of alternants of the ^^th 
order, where A has the following ^ - 2 values : — 
{a\y-^ 
till ' ’ 
] /il 
[«-l 
1 
b 
+ II 
1 
h 
li^ 
1 
h 
/h 
A, 
Zig . 
• • l^n-2 
n-1 
1 
b P. 
. . Z>«-2 
1 
h 
h • 
1 
7^2 . 
• • l^n-2 
and 
: 
+ n 
0 
0 
0 . . 
. 1, h. 
0 
• o 
• o 
.1,7.1 
with similar expressions for B, C., . . . L. 
By adopting the following notation, both series of fractional 
functions may be expressed in a very concise and convenient form. 
Denoting by {abc the sum of the products of a,b,c,... 
taken r at a time, the values of A in the case of alternants of the 
?ith order will, for the latter series of functions, be represented by 
