G14 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
and 
, , r (p — ci)~ + z 2 
^ =1 °g log —jT" 
f=^A=loer ^ v, + ’ cos(|?r+y) 
2 v P 
lr 
IT 
This is to be zero when v:= 0 .'. y=y and 
A 
’ v 7 P 
i (p — «) 2 + ;d . 
lo 8’-^- sm 
w 
9 
But now the distribution of temperature at the other end is given by 
(= 2-Ud° g 
p—«p + : 
leaving only the absolute magnitude A at our disposal. Further, there must be 
supposed a generation of heat all along the circular axis. This example serves to show 
the artificiality of solution given by this form. 
3. For the case of an anchor ring, or tore, it is at once evident that the proper 
functions u, v to take are the well known ones given by 
u-\-vi=log 
p -\- ci -\~zi 
p—a + zi 
viz. : r= const, a series of circles through two points (±a, 0) and it= const, a series of 
circles orthogonal to them, and each containing one of the fixed points. If these be 
made to revolve about the line through the fixed points, we get functions proper for 
two spheres (u ); or the surface formed by the revolution of a circle about a line cutting 
it (v). If they revolve about the axis of 2 , we get functions proper for circular tores 
(u) ; or for two intersecting spheres (v). It will be useful to set down here in a com¬ 
pact form, formulae relating to these functions, which will be required later on. Most 
of them are easily proved and are set down without proof. 
v = 
d + (p + «F 
h lo g pr 
tan -1 1 +tan -1 - 
p + a 
tan 1 
8~*-S +(p _ a) » 
BA r 
0,0 o 
P~ +Z- — CI- 
