270 Proceedings of the Royal Society of Edinburgh. [Sess. 
obtainable from the v th set. A comparison of the results with the equations 
the (/UL + y) th set, he says, gives the noteworthy result 
s=n 
a< £ +v ' > = 2“£ >a S?> 
s = 1 
or 
s=n 
n (m) _ V n (r) n (m-r) 
u ik u si u sk 
This includes, of course, the recurrent law of formation 
S = 1 
if we remember that by implication a$ must be viewed to be the same as 
a si . The variety of expressions for a { S which the identity gives makes 
possible a like variety for 
< ) + < ) + . • • + <>, 
that is, for s m . We may, in fact, put as an equivalent for s m any one of the 
m — 1 expressions got from 
i=n s=n 
2 2 ^ 
i—T s=l 
by taking r= 1, 2, . . . , m — 1. We may even obtain an m th equivalent 
by making r — 0 if we agree to consider a { \ = 0 or 1 according as s is 
different from i or the same as i : in other words, if we agree to place 
before the original set of equations the set 
g°x z = \x Y + 0x 2 + ... + 0x n ' 
g°x 2 = 0x 1 + lx 2 + . . . +0x n 
g°x n = 0x 1 + 0-x 2 -f . . . +lx n 
the truth of which is incontestable. As a consequence the array of s’s 
above given may be replaced by 
22#® 
2 • • 
■ 22«"’ 
2 2 a « a ® 
22« • • 
• 22 a >*~" 
2 
22 a «' . . 
■ 22^ 
22^ 
2 2°®'"°® 
22 ■ ■ 
• 22^r 1, 4r 1 ' 
where s 2 , for example, is represented in the 1st, 2nd, 3rd rows by 
22<«s> 22 a « 
