112 
ME. W. SPOTTISWOODE ON THE CALCULHS OE SYMBOLS. 
Then we have for 
p=M, q=s, 
]( ) 
2)=M— 1, J=S— 1, 
M-i]( ^ 
(m— M) 
ft) \ (m-M + l) /Cb, , /' 
^h,- i 
'll! 
^=M— 2, q=s—2, 
^ \ (m— M+2) 
^=1, q=^s — IVL-j-f? 
W, 1 
'I' 
M 
p=0, ^=s— M, 
m, 0 ](^ 4,,, j* 
the sum of all which will be found, on reference to the expressions for the formation 
of the Os, to be equal to the first term of 0„ viz. and consequently 
cient of vanishes for all values of s not exceeding the greatest value of viz. N M. 
If, however, s is greater than N-M, by any number t, so that s=N-M-j-t, then the 
pairs of values 
p=M, q=s, 
p=M—l, q=s—l, 
are inadmissible, and the pairs 
j9=]vi — ^-j-i, q — s*~^~i“i 
p=M— f, q=s—t, 
p=M.—t—l, q=s—t—l, 
p=0, ^=s— M 
alone remain; and consequently the coefficients of the powers o{ t, for s>N-M. do 
not vanish, and the remainder consists of a series of terms, the index of the highest 
power of w being 
as it should be. i i, ^ i 
As an example, we may calculate by means of the formulae given above, the final 
remainder in the external division of 
?’4(fK+ <P3(?K+ ^o{§) 
