ME. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS. 
109 
Ketuming to the O functions, and writing for convenience the symbolical expression 
p-hm=N— 5, 
where the suffix s indicates the number of units whereby the sum is less than N, 
we have 
‘I>0 = 
4^M^1 = 
4'm<Pn- 
= 
^N- 
<Pn 
4m^2 = 
4^M^N- 
= 
4^11 *?^N- 
= 
^N- 
(13.) 
cPn 0 
and generally, 
0 . . 0 
These formulae give, for the example discussed above, 
^0= <P. 
\(/i + 2\j/2 
(14.) 
= i <p3 
<P2 
<P3 
<P4 
^2 
4^1 +■4'!! 4'o+24^'i+ 4^2 
4^2 4^1 + 24^2 
0 4^,. 
And for the final remainder, the coefficient of 
<Pi 4^1 4 ^o+ 4 'i 24^0+ 4^1' 
^2 4^2 4^14-4^2 4^o+24^'i+ 4^2 
«p 3 0 4^2 4^1 + 24^2 
<P4 0 4^25 
(15.) 
