IVIE. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS. 
107 
and the (r+l)th remainder 
The final remainder is the (N — M+l)th; and the expression will be derived from (10.) 
by replacing r by N — M. 
It remains to develope the terms under the sign of summation in the expressions for 
the Os. In the fii’st place Oo=(pN simply. In the case of O,, the limiting values of^; 
and m are 
p =0, 1, . . N-M, 
m=0, 1, , . M, 
p +m=N— 1. 
These give as the only admissible values 
p=N-M , N-M-1, 
m= M— 1, M , 
and consequently 
In the case of Oj, the only admissible values are 
^=N-M , N-M-1, N-M-2, 
M-2, M-1, M , 
jiving 
(N-M)(N-M-1) 
^2 ?’ n -2 
+ Oi( ^^J_^+(N-M-1) 
+'„)} 
Before proceeding further, it may be well to illustrate these formulae by an example. 
Taking the case of N=4, M=3, we may determine the quotient and last remainder 
of internal division of 
^4(fK + ?>3 (t)'^"+<p2 + (f) 
by 
•4'2(f K+ )’?'+ ), 
and thence the conditions that the latter may be an internal factor of the former. 
By the formulae given above, we have 
^0=?>4, 
which will determine the quotient 
p 2 
