KiRSTiNE Smith 
27 
and 
,r-^ol - ,,_2aJ= ^j^Z^^jl^-. f (2« + 3) {2s + 5) 
(2s + 2r- (33), 
0-2 
which enables us to form „ct^, by successive summations from ^af, = 
Before investigating the curve for „ct;, for a special n we shall first look at „cr- 
for a; = 0 and x = ± 1. 
(9) From (33) we see that when a; = 0 
2^ol is for = 0 most easily evaluated from the formula (13). 
Remembering that in our case nUr = ^ ^, i we find from this 
2r + 1 
1 
2l>^ii TV 1 ' 
and hence by (24) and (25) 
^=,?_ =^=.?_c72j3 5 7 2y + 1 j 2 
(10) To evaluate „a;, for a; = ± 1 we use (32) and (33). The sum in (32) may 
be considered as ^ _ d 1 d Id {x^'-^ {x^ - ly} 
cix 00 cloc ofy doo 
with a number r of differentiations. If these operations are undertaken directly 
upon a;2»'-i (x^ — 1)'' the result is 
a, {x^ - ly + a,_i (x2 - l)*--! + ai (x2 - 1) + ao. 
of which only ao = 2r (2r - 2) 4 . 2 = [r . 2'' 
remains for x= ±\. 
Corresponding to this the sum in (33) comes out from 
^ _ d Id^ d I d {x^'-i {x^ - 1)'-!} 
doCi ijc doo doc oc doc 
by taking (r — 1) differentiations and therefore 
. 2 = |r - 1 . 2'-i. 
.(34). 
^2r-i equals, for x 
= ±1, ( 
2r — 2' 
)(2r-4) ... 
Hence 
a;2 = l 
2r-l 
= ^(4^+1 
and 
'2 
2r-\^,i ~ 
2r-2*^i/ 
or since 
a^l = 
„4=^{l + 3 + 5+ (2,^+1)}, 
n<yl=j^{n+iy (35). 
