34 
ME. W. SPOTTISWOODE OX AX EXTEXDED FOEM OF 
in which all terms will vanish for which 
m<0, and 
In order to determine the effect of the operation 
d 
on a given function, we may proceed, as before, by making 
or 
[(u.u, . . ■■ 
(A. A, . . I*y)"= (u.u, . . !)"(«.«. . • HxyY ; 
and then the coefficients .. will be given by the system, of which the follovring is a 
type : 
A;=[Moao,,„Pi 
+ (Ml ao_ _ 1 Pi+ 1 + Ml ai,„ P,-) £*■ 
I • • • 1 
as above. In order to solve these equations for a . , we may regard the expression 
A 
within the brackets [ ] as a function of ; and resohing it into its factors, we may pro- 
ceed by way of operations instead of direct elimination. Let^;!,^^,? ke the roots; 
then 
Ai=a„,o Pi+„i(£‘'' — — Pa) •• )(li- 
and 
But 
A, 
“m, 0 A i + ni 
-Pl)- 
A,: 
■-h+s 
p\ p. 
A A 
£di ^ di 
Pi+m PiV ■ ih ■ K 
1 Aj+i 
Ai 
J Pi+m 
= _i^_A4 
pA^m p, P 
i+m+l 
+ ••4 
Py~*i\+m+lJ 
Pl) 
-^Ai 
Pi 
i+m 
A, I 1 Aj+i 
_i A 
r-'Vn+m + lJ 
Pi+m P\ Pi+»n+l 
I 1 / A;+l 1 Ai.|.2 1 ^ \ 
PjVP' + m + l Pi Pi+m+2 P„ + „, + i/ 
= (-) = 
+ 
1 
fA, 
1 ^ 
\ 4- 
/I, 
A„ 1 
1 P ' 
\P. 
Pa/ 
/P.+..+i^-- 
\Pi 
' P«+.+.J 
