ME. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS. 
105 
and in this expression 
while the general term of the (p-series, i. e. the coefficient of will be 
/_v-, ( »(»-!) ••(»-?•+ .jr _ w(ra-l)..(»-y+l) 
^ 1 1.2. .r 1.2..(r + l) ^'•+* 1.2..r 
— '' ' \.2..r \ /•+! ’■+*) 
_ (w + l)»(n-l)..(»-r+l) 
^ 1.2..(r+l) ^>-+1’ 
which proves the general case ; so that generally 
E.„=E.+<p<A7>+ . ,, 
■ ('‘•) 
the upper or lower sign being taken according as (w+1) is even or odd; where E,„+i is 
the remainder after external division 0„+i(f)7r"'^'+<p„(g’)7r”+ ..(p^ by 4'i(^)^-i~'4'o(^)‘ 
For the quotients Q„ Q 2 , . . we have immediately 
Q,= i|<p,.+E.Q|, 
+ E 1 ^ + E^ ^ I . 
Q.=^{-!1X-+E.("„ . . E._, Q|. 
This completes the solution of the problem of division by a linear factor, both 
internal and external. 
§ 3. To divide internally by 
The first term in the quotient will obviously be 
‘Pn n-m 
( 1 -) 
and the product of this into the divisor may, by means of Leibnitz’s theorem, be written 
thus : 
pL ( 2 .) 
Ym 
where means the result of the operation or alone, and 
MDCCCLXII. 
P 
