190 
Proceedings of the Royal Society 
Hence 
V n =- 
n 
^2 ( n even), 
+ n - 2 ^ odd )’ 
that is 
U ft -wU n . 1 ~ 
n TT ( IP -1 d 
or 
U 71 -?iU n _ 1 + 
71 jj + / _ n«-i d 
n -2 Vn - 2 1 < L) n- 2 ’ 
(5). 
From this the successive values of U M are got with ease. 
3. To obtain the generating function of U we return to (4), and 
write it in the form 
u n = {ii - + U M _ ! 4“ 2U n _ 2 ~{n~ 3)U»j_ 3 + U n _ 3 — U,j_ 4 , 
so that if u, a function of a?, he the generating function, its differ- 
ential equation is at once seen to be of the form 
_ du du 
u = x z ^ + xu + zx l u - a? 4 ^ + xhi - a?% + </>(a?) . 
By trial, however, <f>(x) is readily found to be x 5 - 2a? 4 + a? 3 , conse- 
quently the equation is 
du 
(a; 4 - a? 2 )^_ + (# 4 - a? 3 - 2a? 2 - a? + 1 )u - a? 5 - 2a? 4 + a? 3 . 
Integrating in the usual way we first find 
/ 
h? 4 - a? 3 - 2a? 2 - a? + 1 , 1 , a? 2 — 1 
dx = x + - - log 
tAj vU 
and 
u 
1 . x 2 ~l 
a? + - + log 
rv* ^ rv* 
t/U t/w 
1 X 2 - 1 
exp\ a? + - - log — 
/y» ^ ry* 
vL/ tAs 
= exp^ 
'S\ 
={x-\ y( a ’ + ?y'/ + 4 
) 
x 
X 
a? 
a? 5 - 2a? 4 -fa? 3 ) 7 
f dx. 
} 
x l 
/y>4 __ /y*2 
vO vU 
a? 3 (a? - l) 2 
x 2 - 1 X a? 2 (a? 2 - 1) 
dx. 
dx 
(x + 1) 2 
This does not really differ from Professor Cayley’s result (Proc. 
E.S.E. 1876-7). The apparent difference is due to the fact that in 
the one case u is assumed to be of the form U 3 a? + U 4 a? 2 + . . . , and 
in the other of the form U 3 a? 3 + U 4 a? 4 .... 
