S IrH 
f 
14 
Similarly, 
And, generally. 
ME. W. SPOTTISWOODE ON AN EXTENDED FOEM OF 
=(V-2)S^ 
=VHH,-2HS, =(V-2)HH„ 
SS?=VH?-2Hf =(V-2)S?, 
W3 r7 OWW r7W2 OW^ 
►-wi — Vi— '1 — ^<-<*-<1 — V*i — 
ssr'=(v— w+i)Hrs 
H”= ViSr ‘ 
Before developing these expressions, the following formulae deserve notice 
(S+H0^=V^+2VV:+V?-2VT2Vi 
=(V±Vi-2XV±Vi). 
Similarly, by arranging the terms of the third order thus. 
we have 
And generally. 
>^3 
VS" 
-2S", 
2VSS. 
>+ V.S" 
-4S3.-23S 
VS? 
+2ViSS, 
OW2 9W2 

>— fj 
v,s? 
0 w w 
(H±H.X=(V±Vi-4XV±V.-2)(V±V0. 
VS”-' —{n—l) H”-' 
^ VS”-=*Hi+ ViS”-' — (w— 1)^^ — (w— 1) 
l)(ra- 2) (ry, u(^Z: 1 K!LZ^w»- 3W2 
1.2 " 1.2 ^ Vi« -1 — 1.2 "" 1.2 - -1 1 - 
Taking the sum of these, the coefficient of 3”-^-'S? on the right-hand side will be 
{n-\){n-2)..{n-p) (n- l)(w-2).. (n-;?) , (n-l)(n-2) ..(«-;> + 1) 
1727:^ (V+V, )-(«-!) I7277P (n-p) i.a..(p_i) 
1.2..(p + 1) 
\.2..p 
(w — l)(n— 2). . {n—p) 
' 1 . 2 . . jo 
{v+V.-2(?i-l)}. 
