116 
ME. W. SPOTTISWOODE ON THE CALCUIiUS OE SYMBOLS. 
And generally the expression for the ^th remainder may be written 
M 
0° — s)'lM(7r— s — 1) ..I/mItt— s— ^ + 1) 
<Pn-/+ 1 (’T — s) ^^M(7r — s) 
0 
4^M-«+i( 7J' — s) 
( 10 . 
(PnCtt — S — ^+1) 0 
The last remainder is the (N — M4-l)th. Then 
^=N-M+1, 
N-s-^+1=N-5-N+M-1 + 1=: Ms 
N-^+l=N -N+M-1 + 1= M 
M-^+lz=M _N+M-1 + 1=2M-N, 
and the remainder in question 
= M — s 
^S=:0 5 
^ — ^"Tl)- 
X 
'IftdTT— s)vI/M(7r — s— 1) — s — N + M) 
(11.) 
^n(‘^ — — N-f-M) 0 . . •4 ^m(^ — ^ — N+M), 
in which the coefficient of vanishes, as it should. The last term, -viz. that independent 
off, 
(12.) 
X 
vI/jil’T — M)\I/M(7r — M — 1) . .I/mItt— N) 
<Po(7r) ypoi^r) . . 0 
(pM(7r — M) 4 m(7!’ — M) . . \|/2M-n(’>^ — M) 
<PN(7r— N) 0 .. ■4 /m(’J'— N); 
and if N=l, the result agrees with that given by Mr. Hussell*. 
§ 6. To divide XZ^f<PnM externally hy 2”:“f“4™(7r). 
The first term of the quotient will be 
<PnW 
The first remainder 
^ 4/m(’1' + N — M)‘ 
y"=^p«(n tTri ;y”»=M,N-M+OT + ^ / \ 
( 1 -) 
4/m(w + N — M) 
= / N 4'M-s(7r + N— M) ^ , 
* Philosophical Transactions, vol. cli. p. 72 
( 2 .) 
