j\IE. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS. Ill 
and if we make this expression =02^3-’^“^, the next remainder will be 
the first term of which is 
1- /d) \ m — M + 3 /d) \ m — M + 2 ■ /d) \»k — M+It"! 
r, , /toX /„ /'fl’oV (M-1)(JI-2) / 4 .„Y' M(M-)(M-2) 
1.2 
and generally the (r+l)th term in the quotient will be 
and the (r+l)th remainder 
<1> 
^ _ N - M-r 
4 'm’^ 
/d) \ (.m-p) 
The formation of the Os is as follows : — 
^0— ‘pN? 
\ (w — M4* 1) 
«-!](£ +k ) 
Cp \(m-M + l) 
\ (?7i — M + 2) 
f /<i) \ (m— M + l) /<J) \ (m — M + 2) 
) +[«, ) 
+ KM-3](5i) }, 
+ -K1 ](^k-) +KO](i^r) }• 
The final remainder is given by the formula 
( d) \ 
and the general term of this is to be found as follows : 
N— ^=N— s, 
p — §^=M — 5. 
e. 
(9.) 
( 10 .) 
■ ( 11 .) 
