626 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
clY 
which on reduction and substitution for — becomes 
E(A / ) = ^—^7 ( 2 E' —& 3 F') 
and 
•• -Q 1 v /2= v/ 2 ^{4E- 2 ^F ~ 2 ( 1 +/^)F'} 
2 y /9 
Q« = 2 
s/k 
(2n — 2) . . . 2 [ «, 
f(E-F) 
'** "(2«-l)...3l y/l 
We have in fact proved that 
dd 
(cosh n + sinh ?t.cosh 6) 
5±T=(- 1 )V af 
» 2 J n 
COS 
0 y/cosh w + cos 6 
By means of the identity 
or 
EF'+E'F —FF'=^ 
1 hr \ 
E-F=EQ_ ef 
Q„ can be expressed in the following manner, viz.: 
_ (2a —2) . . . 2 f -7r«» _F7 «„E ± „ 
Q"- 2 ( 2?l _l) . . . 3[2F y/k’ F ( vX //7 +£ - ) ; '- ]F 
9. The following relations between P and Q functions will be useful in applications, 
viz.: 
(a) P«+iQ« P/iQ/i+i — 
9. 
7T 
2w +1 
7T 
(/3) P'»Q«—P»Q'«= S - 
(y) PhQh+i—P / «+iQ'«=(2?i+i) y 
(24) 
They are easily proved, for substituting for P„ +1 , Q „ +1 from their sequence equations 
it follows that 
