32 
MR. W. SPOTTISWOODB OX AX EXTEXDED FORM OF 
Then changing inB, into a^_p4'/3m-p2'^S and compensating the change by 
writing (^ + l) for i in the numerical factors to the right, we have 
A=K . . (^— ^+2), 
B=K ?■— 1) .. .. , 
the factor s*’ having been added in order to indicate that ^ must be changed into (2 + 1) 
in the expression upon which B will presently be made to operate. Then making 
= 2[KH I (^+l)a^«i+i +(2— } 
+KH,„N^ (^+l){(w— m— +(w-?)/3^«.}]a’'‘-'y 
=2 
KH { (^ + 1 )[(w — ^) 1 + (^2 — ^ — m - 1 ) Ja,„ j 
Now 
and 
Hence 
C=:2 
= 2 
= 2 
= »Np) — (m + 1) 
(i— m)„JN^_l + (^+l)„^Np 
= (*+!)( mNp-l+ mN^)~Oj^ + l)mNp-l 
= (« + l)m+lNp— + 
= (^+l— 
.,N^KH{(w-^-»^+^-l)(^+l)a,„«,^J+(?^-^)(^-^+l)/3„«,} 
m+ 
ff 
m+ 
,Ny(a+^2 *■) •• ''0(«m-p+i£'^'+/3,„_p+,) .. (a„6*+/3„) 
(n—i) .. {n—i—m+p))i,. (^— 
which proves the formula generally. 
This expression may, however, be transformed into a more convenient shape, as fol- 
lows. Let 
5, Sa . . = (a ai . . 
(«i + /3,£ '^0(“2+/^2£ '^i)=(aa, .. a,„Xl £ 
then 
