of Edinburgh, Session 1882 - 83 . 
239 
Now assume 
^.m -1 
'(■n-m + 1 • 
hfi - w - 2 IPm • b ,1 - 2m+2) • ^ 
+ ilh • bn-m + i^4 • ^^n-m-1 +2^5 • + • • • + Pm • -^M-aiH+s) • 
+ ijK • hn-m-^2^r\+\' ^^n-m-1 +2^r + 2 • ^^n-m-2+ • • • +i^»f ^^^-2^« + r) 
+ (2^m-l • K-m-^V^n • ^ 
”1“ P»t • j 
operating on this equation in like manner it may be finally shown, 
since by the lemma 
Pi • hfi _ 4- 1 ”b 2^2 ’ b}i - »w “h Ps ' bfi - m - 1 ”h . t . ”t" pm • b 2^ -fa — br,i _ ^ 2 
that 
^n+l _ 7; /^m-1 
it/ — ' ' « - jn-f 2 “ 
+ (P2 • -^^n-m,-fl+2^3* + + - * • +P^ • ^^n-em-fs) • 
+ (P3- ^^«-m + l + P4- ^bi-m+2^5* 1 + * * • ^ Pm • ^?«-2m-f 4 ) • 
+ iVx-' ^^«-w-fl +2^r-fl • bn-m-^Pr-^^' + • *' + Pm ' ^^n-2m + r -f l) ' 
+ (Pm-l- ^br-m-fl+P^* ^ 
+ Pm • • 
Thus the proposition is completely established. 
3. The following extensions of the foicgoing lemma, particularly 
ill the case of a quadratic, may be worthy ci notice. 
If a, j8 be roots of = px-^q^, then 
bn = V . . . (1), 
where is the sum of tlie homogeneous products of a, S 
dimensions. 
Kepeating the principle contained in (1) we find 
=2)2 ^ + 22)2 • bn_^ + (f . 
^(P + 1?) . symbolically . . (2). 
VOL. XII. 
Q 
