ME. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS, 
the first term of which is 
117 
l'M-i(7r + N — M) ^ 
whence the second term of the quotient will be 
1 
- / N \1m-i(’!-+N — Al) ^ ^ ,1 
+ 4/M(’r + N-Al) 
1 
+ N — M — + N — Al) 
TT-fN-AI) 
*?N (tt) xf/M (tT-I-N M). 
( 3 .) 
Similarly, the second remainder will be 
:..-j = N-l „„ / ^ N-M + m-1 + N AI — 1 ) 
'l M (’'■ + N — AI — 1 )rI/M (”■ + N — AI ) 
<pN-i(7r) 4/M_.,(7r + N — M) I 
<P^(7r) (tt + N — M)i 
'y*^^UN-s-i ^ \ 4 ^m-s(7!' + N AI 1) 
— I^N-«->V’^>'~vJ/M(7r + N-AI-l)vJ/ji(7r+N-AI) 
^s=m 
“'s=of ^ M — l)\I/_\x(7r + N — AI) 
<pN-i(7r) x|/M-i(7r + N — AI) I 
‘PnW 4'm(’I' + N — AI) ! 
<Pn-s-iM '^m-.(7J-+N— AI— 1) 0 
<Pn-i W (tt + N — AI — 1)^}/M_,(7r + N — AI) 
<Pn (tt) 0 4^11 (tt + N- AI) 
K4.) 
The rth remainder 
N_«_«+i i ^ 
— 's^o ? + N — AI — / + + N — AI — i + 2 ) . . + N — AI) 
X 
^An-s-^+iC’’") ^+1) .. 0 
<pN-i+i('^) 4^m('^+N— AI — ^-j-l) .. 4 ^m-^+i(’*'+N— AI) 
( 5 .) 
CPnC’^') <1 
and the last remainder, viz. the (N — AI + l)th, 
4/m(^+N-AI); 
— A,=o e 
l'.\r(’r)'I'M(’'’+l) •• l'ir(’r + N — AI) 
X (pM-sM • • 
(6.) 
<PmW •• 4^2M-n(’!‘+N — AI) 
0 • • W’^'+N— AI). 
In the case considered by Air. Russell, viz. AI = 1, (6.) gives only the single term 
{\I/,(7r)4/,(T + l) . . •v|/,(’r + N— 1)} ’ 
<PoM • • 0 
<p,(7r) 4/.(7r) . . 0 
<Pn-iM 0 .. 4/o(7r + N — 1) 
<Pn(^) 0 .. 4/, (tt+N — 1), 
