ME. W. H. L. ETJSSELL ON THE CALCTJLUS OE EUNCTIONS. 267 
where 
Let 
/(^)=f|^+7 cos cos sin cos ®)| 
sin cos ^(£®— £-®)|. 
Next let the functional equation be 
n , 2 
*=-rr+ 
and the equation will be transformed into 
this may be written 
Let 
(p,{^z)—a(p,{z)=^x{z)- 
1 
z= 
30+1 
whence 
“^(3^0 +“’^^(3^} "I"”' 
The same method evidently applies, and we thus obtain the value of ‘p(x). 
In the same way we may treat the equation 
(p(a„-\-a^x-\-a^+a^^+ . . . -\-ajx”')—cc<p(x)='F{x), 
where a^, a,, a^,... must be supposed given in terms of a^. 
We will now consider the equation 
{n — r)r'^ +2rn^ + 2r {rr^ — nt^ — 2nn')x + {{n — r)r'^ + 2rn'^) x^) , -p,^ % 
[tI — r’)r^+2r’n^-\-2j^{rr^ — rlr — 2nv!)x-{- {{r^ — Ci<p{^X) [W), 
n +r cos z 
W — / ~1 • 
n' + r cos^ 
Then the equation becomes 
( 7i + r cos 2z 
«' + / cos 2z 
<P 
Let 
| — a^|- 
w + rcos^l n + rcos^r'l 
Ti' + r'cos^'j Li' + r' cos 5'j’ 
which may be treated as before ; as we may write it 
^,(2z)—oi^,{z)=x(z). 
Similar methods will evidently apply to the equation 
