MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
G19 
In (14) put 
with 
then 
or 
where 
and 
It is clear from this that u n is of the form a. n u x —j3 u u 0 where a„./3„ are rational integral 
algebraical functions of 2C; a n of degree n— 1, and of degree n — 2. The first 
three values are (writing 2C=x) 
u 0 =u Q 
«i= u x 
v^—xu x -\u^ 
l h — (x 2 —c l )u 1 —^xu 0 
We can now show that a n , (3„ are of the form 
a*=+ a n , x x n ~ ?J J r a ll oX n ~ 5 + . . . d -a nj x n ~" r ~ l -\- . . . 
For supposing a a of this form, i.e., wanting every other power of x, it follows at once 
that a n+1 is of the same form, and it is seen above that a 3 is of this form, whence the 
statement is generally true, and so also for /3 U . 
Now a /t satisfies the equation 
OL n - XCL n _ x ......... ( 15 ) 
with 
a 0 =0 a x = l 
Hence substituting the above value for a n we must have 
Mn.r — tt'n—i.r o.r— \ • 
Ct >i. 0 —• • • — f 
4 L 2 
_ (2n — 2)(2n — 4 )... 2 
n ~ (2%-l)(2%-3) ... 3. I n 
P o=u Q , P y=Ui 
v _o CV 
M »+2 ZUf ' i + 1 T 272.(2% + 2) W " —U 
— _C?G/+i ^ ii^n 
(2» + l)» (272.+IF j/JL_L_ 
* 2%(2?l + 2) (272+1)2-1 “ t " 2 V 2 « 272 + 2 
r — 1 
l U— 9 . 
also 
(1G) 
