294 Proceedings of the Royal Society of Edinburgh. [Sess. 
-J**p + **£ 1 + etc. + Y 1 + “i + .2? + etc. + ) + n(n+ 1)(^ + *%'■+'■ ^ + etc. Y 
\ X 4 X 6 X 8 /\ X 2 X 4 / \£ 3 X 5 X 7 J 
i + ^ + ^ + «_s +e tc.r 2 
X* X 4 X b / 
2.3.cqw 
1 + ~i + -f + etc - 
n+1 ' 
If we now choose n in the first form, so that (n-\-2) = icn J and equate the 
numerator to — yL, we obtain G>'' K (oc) = — [©^Jp/^ 4 - And the coefficients 
oc 
a v a 2 , a s , etc. can be determined from the following equations : — 
2.3.cq = - ; 4.5.a 2 = (/3.1 2 — 2.3)a 1 2 ; 6.7.a 3 = (2/U.2. - 2.3. - 4.5.) ai a 2 ; 
n 
r—- 
2 
1 
r = 1 
2t(2t +!)«,= g {2^3.r(i-r)-2r(2r + l)-(2j-2r)(2*-2r+l)}a r a^ r 
( / j\2 \ 
+ | !) j <4 (* even) ; I _ ( i.j) 
2*(2« + 1)0,= g { 2 p.r(i - r) - 2r(2r + 1) - (2t - 2r)(2t - 2 r + l)}«^ r (< odd) ; 
r=i 
where 
/3 = 4(»+l) = ^±Y). 
(k-1 
Similarly, if we choose n so that (n + l) = icn, and then choose oq so that 
2.S.a r n = l, the first term of the second form of ®" K (x) becomes — 
and the second term, equated to zero, gives us the following equations to 
determine a 2 ,a 3 , , etc. : — 
4.5.o 2 = /V ; 6.7.a 3 = (2/3.1. 2 - 4.5)a l( i 2 ; S.9.a 4 = (2/3. 1.3. - 6.7)aja 3 + (/3.2 2 - 4.5)a 2 2 • 
’'“IT 
2i(2i+l)a < ={2/3.1.(7- l)-(2i-2)(2*--l)}a 1 a». 1 + g 2 (2/3.r(i-r)-2r(2r + l) 
r—2 
— (2i — 2r)(2* - 2r+ l)}a,.a ( _, + j /3^ —i(i+\) j a; (i even) ; 
•••( 15 ) 
J— 1 
2 
2i(2i + lK={2/3.1.(i-l)-(2i-2)(2*-l)}a 1 a i _ 1 + g {2/3.r(i - r) - 2r(2r + 1) 
(2* - 2r)(2i - 2r + l)||i r a i _ r (* odd) ; where /3 = 4(»»+ 1) = 
4 k 
(k - 1)' 
