106 
MR. W. SPOTTISWOODE ON THE CALCTJLTJS OE SYMBOLS. 
Then the remainder after subtraction from the dividend may be written thus : 
J 
^m+p = 0 
^p=o 
ip = N— M ^7n=M 
• • • (3.) 
since the coefficient of vanishes. With a view to the second term in the dividend, 
the first term of the remainder (3.) is 
t •r.ip=N— M •^m = M 
. . . (4.) 
in which the limits of p and m are subject to the further condition ^+w=N—l. The 
terms under the sign of summation will be evaluated hereafter. Putting the expression 
and, for the sake of symmetry. 
(5.) 
< Pn =<>0 
( 6 .) 
‘hi 
the first and second terms in the quotient will be ^ and respectively; 
and, in the same manner as (2.), the product of the second term of the quotient into 
the divisor may be written thus. 
( 7 .) 
and the remainder thus : 
^m+p=:N— 2 
^m+^ = 0 
( 8 .) 
But since, when Pi = N — M, [N — M — l,^j] = 0, we may, without altering the value of 
(8.), change the superior limit of from N— M— 1, to N— M; and by this means we 
may write the remainder (8.) in the following form: 
-ym-hp = N — 2 
^m + |) = 0 
Similarly, calling the first term of (9.) the third term in the quotient will be 
d>, 
2 .yN-M 2 ^ corresponding remainder 
1 
M 
+02[N— M— ; 
and so generally the (r-f-ljth term in the quotient will be where 
T IVT 
( 10 .) 
( 11 .) 
