MR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
625 
/. (2 n —l)Q«_ 1 +(2w+l)Q*+ 1 — (— ) /Hl c[ 
J i 
= 2 (—)" c j To 
— Id — cos n+ 16 
0 (C + cos dy 
n sin n6 sin 6 
o (C + cos 6) 
d 9 
=4wCQ„. 
The same sequence equation as before. Hence it is only necessary further to show 
that Q 0 , Q : are the same in the two cases. 
Now 
tie 
when 
Qov / 2 = f 
J n 
0 \/C + cos 6 
'** cW 
aJ C+l — 2sin s | 
cie 
\/C + lj o \/l —A" sin~ 6 
\' 2 = 
C + l 
If C, S be eliminated between and & 2 =2S/(C+S) there result the equations 
h 
,_2+\ 
1 + A 5 
Jc'= 
1 —X 
1 + \ 
Hence by the second quadric transformation 
Fv=(l+*')F' 
and since 
Again 
Now 
X '—2 k 1 /(1 + Id) 
Q 0 =2 v /FF / . 
■Qi-v/2=XV2{ a 
cos 26dd 
+1 — A/ 2 sin 2 6 
=^{2E A ,-(l+X 2 )¥ v i 
A. 
E(X')=W^2+A?E(\') 
=+'f;'lM4+>F'+(l+0F' 
4 M 
MDCCCLXXXI. 
