of Edinburgh, Session 1882-83. 
243 
Finally, in the case of the equation 
+ . . . +p>m ) 
since 
K==Pl-K_^+P^. A^_2+i93. /i„-3 + . • 
it will follow in like manner that 
A» = {Pl +1>2 +P3 + --- +P«f ■ 2 
=( 
=( 
and generally 
= (Pi +P2 +Ps+'--+pT- will > 
the number of different h’s being (m--l)r+l, which are repeated 
by the same law in going from the beginning and the end, and such 
that the sum of the number of all their coefficients is equal to the 
sum of the series, 
m-\ (m-l)m (m-l)m{in + l) 
1:2 + 47273 
(r + 1) (r + 2) (?• + 3) . . . {r + m - 2) 
1.2.3... (792-2) ’ 
that is 
(r + 1 ) (?* + 2) (r + 3) . . . (r 4 - m - 1 ) 
172.3... (m-1) ’ 
which is the number of terms in 
(pi+p2+^^3+* • • 
