MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
645 
The expansion for z has already been given, viz. : (for points not on the axis) 
Hx/C — c sin nv 
To determine /x we notice that at the critical circle (as everywhere on the plane 
of xij) 
d<}} dv 
dv dn 
Taking a point outside the critical circle 
-y =/ x 
= -v 
H-Qn 
The easiest way to calculate this is to make the point approach the critical circle, 
i.e., 11 = <x >, when 
-V=Dim (C— l) ! Qj 
=P im ( C -tf (C + Sc^ 
/* 
dd 
L, 
a J 2 ? cosh 3 - a 4: y/'2, 
which gives the theorem 
Hence 
tn z Q lt = i= f^ cosech 3 
4«a / 2 
^ v 
-V x/C— cS(A„P /J sin —a,,) — nQ /t sin ny) 
£ i 
where y =0 when it=u 0 for all values of v. The terms in cos nv would merely 
increase <f> by the series for a constant, we may therefore without loss of generality 
put ot n = 0, and then, using dashed letters to denote differential coefficients, 
1 7 r bcp 
=t 
4y/2 aY Sit [2-v/C 
; (A„P ;< — nQ») + v/C-c(AJPk— toQ'*) \ sin ra; 
= 0 when u=u n 
S{S(A«P„—nQ„) + 2(C—c)(A»P'»—sin%y=0 
A„ +1 Pk +1 +Ak_ 1 P' / _ 1 -A /i (SP„+2CPk) 
= (n + 1)Q „ + i + (w— 1)Q — «(SQ„+2CQ „ 
