644 
MR, W. M. HICKS ON TOROIDAL FUNCTIONS. 
and the condition is that when u=u 0 </> 1 +</> 2 = 0 
. . a,= v, A ,r-„=- 4,lbo ' 
Whence 
a A =if, A,F\,= - J ~T'nQ° n , and A 0 P» 0 
. p' QO 
7r 
^=:^ v /crra{2^(F s Q,-Q»,P„) cos »(«-»')+^(I«oQu-Q ,j oPo) 
77 l 1 » 1 0 
. (41) 
and the general solution when the tore is insulated and has a charge of its own is 
found by adding the potential found in the last article. 
Also if the section of the wire be very small we can find the capacity of the system 
approximately, by supposing the wire to coincide with one of the equipotential surfaces 
near it. 
19. As an example of the use of tesseral functions with constant surface conditions, 
r we will take the problem of the electrical induction on a tore under the influence of a 
point arbitrarily placed. We lose no generality by supposing it in the plane of (xz) ; 
let then its co-ordinates be (u'.v'.O). The potential due to this for points within n’ 
has been found at the end of Section III., viz., 
(f)—~ v/fU-cXC'-c 7 ) SL„,„P',,«Q^ cos mw cos n(v-v') 
As before, the potential of the induced charge will be of the form 
<f>= \/C — c 2A Wf »P w .» cos mw cos n(v—v) 
and (the tore being u 0 ) 
/Tv / 
\ po __v L r. P' no 
rs ~/dji 1 ' nisi — / x “ mji 
/ 7 n 77777 777 P' 
$=11 
— " X ’* ' 1 6 -2L m ,n —(P°«.«Q». rt —Q°, COS MIV COS 1l(v— v) . 
(42) 
When the point is on the axis, all these terms vanish (§ 14) except for m = 0. 
If necessary, also, the capacity of a tore and a very small sphere can be found 
approximately from these formuke. 
20 . One more example illustrating the application to cases of differential surface 
conditions may be given. Take the case of a tore moving parallel to its axis through 
an infinite fluid with velocity V. Here the conditions are that if c f> be the velocity 
potential for fluid moving past it, 
<?*)=— Va-f ^ 
and 
b<f> , 
<— =0 when u~u n . 
OIL 
