ME. W. SPOTTISWOODE ON THE CALCULUS OF SYMBOLS. 
And it is not difficult to see that the first term of the rth remainder will be 
1 
115 
„N-r . 
•■I'M (tt — 1 ) 'I'M (’>■ — 2) . . (tt — r-) 
(PN-r+l(’!'— 1) (tT—I).. 0 
( 6 .) 
in which determinant every column after the first consists of only two terms, viz. 
\ly_,(7r— .s) and \|/M(7r— s — 1). Hence also the (r+l)th term of the quotient will be 
1 
„N-M-r_ 
'1'm(’*’ — M) — M — 1 ) . — M — r 
<PN-r(7r — M) • • 0 
<PN-.+ i(’r— M — l)4'M(’r-l) • • 0 
( 7 .) 
<pN(7r— r) 0 . . %J/M(7r— r). 
As to the other terms, than the first, of the various remainders. In the first 
remainder, the first term of which is given by (3.), the (s+l)th term will be found by 
making « = N — -s, = M — s, in the expression 
<pn(— m+™). 
which gives 
1 
I'M (’!•-. 9)! ‘Pn-sW 'f^M-X'^) 
i — «) — S). 
Hence the entire first remainder may be expressed thus : 
<Pn-sM 
Similarly, the general expression for the second remainder is 
'I'mC’T — s) 
(8.) 
2:„=o ^"*=0 f"" M + w) 
!<PN(’r— M-j-m— 1 ) (’t— M+w^— 1). 
which may be transformed thus : 
n=N—s, m=M—s, 
•V'» = M.N-«-1 f 'I'M-»(^) 
■1) 
■2,=o f 
I' M (’T — S) 'I'M (^ — S — 1 ) 
<Pn(^— 5 — 1) S — 1) 
<Pn-s-i(^ ) ) 0 
<PN-1 (■TT — S ) \pM (‘^ — ^) — S ) 
(t—s—1) 0 4/m (^— S— 1). 
Q 2 
( 9 .) 
