8 THE REV. F. H. JACKSON ON 
The terms involving «'"~*” give us 
a\ion = rrt2 [2x — 27 — 2]! n—27r—1p[n—271] 
ee ae SEO 
_ Kp( _ 1)""[n =. liga [2n —UnS 2]! 
[m—7- 1]! [nm — 27]! [7 — 1]! (2)n (2) pa 
rn” —2rg{n—2r] 
These may be written 
ijn [2n —2r]} scans sea a ae 
i ae - 7]! [nm — 27]! [7]!(2), (2), (p — 1)"[2n - 2r - 1] 
Putting « =p” the large bracket reduces to 
(p” = Nig ze 1) 
(p — 1)°[2n - 27 - 1] 
which is 
[7] 
so that the coefficient of a!"-2" is 
= Pprr+e [2m = 2r|! —2r 
lana Oe. 
and the series is 
[72 |Pin(wA) 
establishing 
[m]Pp(@A) = ALQ2n— 1]ePp_(w?A) — pH [n — 1]Pinr_o (aa). . (45) 
9. 
Another property of the function is 
2. EP iny(@PA) ? TD inn(@?A) 2 > p p) 
Nae d(x") eee dae) =X [7] ' a] in X) a; TP in-u(®r) } . (46) 
By means of this we establish that 
i a wy 
P,,(l-p?) = p?P,,y(1-p?) . : ' > 4 
and if 7 be integral 
Prai(lp?) = p2 : : ee (5) 
which corresponds to the theorem that the sum of the coefficients in LEGENDRE’S series 
is equal to unity. The proof of (46) is as follows : 
5 or! 
Pal) = LOD a 
therefore 
d oP = = — 1)%pr-7+2y 2-2 n — Ir]! | pp'in—2r-1 
d(c) Font”) sare [][n— te 2 Tt tai 
Similarly, 
d { . Ss aaa ; [7 -- 2r+2]!- [nm - Qr+ 2] 
—r - LP = = r—1jyr—-Lr+1\n—2r+2__, eels = n—2r+l] 
fe »| Be aA =F eT R=a ea ee 
