634 
MR. W. M. HICKS OX TOROIDAL FUNCTIONS. 
But (32) 
(2m — 1)SP*. 0 = — (2m—3)SP*_ 2 . 0 —4 (m— 1)CP„,_ L0 
Therefore the above 
— b {Pl.oQo.0 Po.oQl.ol 
But from the first of (32a) 
S(Pi.oQo.o P0.0Q1.0) — P0.1Q0.0 Pq.oQo. 1~ - 7r 
Hence 
and 
p n _ p n _..n (2n-l-2m)(2n-3 — 2m) . . . (1-2 m) ^ 
1 {2n + x + 2m )(2n-l + 2m) . . . (l + 2m) ,7T 
p/ pv _p ry _ (2a 1 2 T/i) ... (1 2 rn) * 7r 
Ml, iu^Xjiil .11 Til.il / q 1 I 1 O \ /1iO \ ci 
(2n+l + 2m) . . . (1 + 2 m) S 
• ■ (33) 
In the same way, or by substituting in PV«Q/«.«—P^QV# the first of (26) or the 
first of (31), there follows 
-iQ* 
' P — ^(P/« + i.rfQw.« P j».»Q«»+i.«) 
14. From the formulae now developed it is possible to find the complete integral of 
the general differential equation. But as in applications the co-efficients are deter¬ 
mined in terms of definite integrals it will be well also to consider the solutions from a 
different point of view. If any potential function be expanded in a series of multiple 
sines and cosines of v, w, multiplied by vC — c, we know that the co-efficients must 
be of the form AP+BQ. Now such a function is the inverse distance of any point 
from a fixed point. Let us choose as fixed point, to simplify the expression as much 
as possible, a point on the axis of x within the critical circle, say ( u'.tt.O ). Then the 
distance of ( u.v.w ) from this is 
p ( trA+W ccr + c - ss ' cos ^ 1 
Hence if — p.p. be expanded in a series, the coefficient of cos mw cos nv 
ViCC+c — SS cos w} 
will be of the form AP /Si „ + BQ „> n . Further, for points within the tore u' (i.e., u>u') 
A = 0, whilst for points without, and therefore including the axis (u < a ), B = 0. 
Hence 
/o+t(T 
J o •'0 
cos mvj cos nvdvxlv 
•y/{CC + COS v — SS' cos w] 
AP,„., t or BQ,„„ 
according as u^u'. 
Now r if the fixed point be on the critical circle B is always equal to zero and 
(C' = S'=co) 
