MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
G15 
. e’ l+vl + l ^ 
pJ rZ i_a - i+vi _ 1 
sinh u , 
0 nnali m — r>na 
z—a 
cosh u — cos v 
sin v 
cosli u — cos v 
(5) 
dv, _ g _cosh u — cos v _sinh u 
dn a p 
whence the statement made above that p£=f(u). 
Let It, r be the radii of the circular axis and normal section of a tore (it ); r the 
radius of a sphere ( v ); then 
R O 0 O 
cosli u=~ 
sinh u— 
i/R 2 —r 2 « f" 
Sill v = 
(6) 
Further, if r, the radius of a tore to a point P (a, v) make an angle 0 with the plane 
of the ring 
cos 
r — R cos 0 
R —r cos 6 
sin v— 
\/R 3 —r 3 sin 0 
R — r cos 0 
With the above values of (it, v) the general equation for toroidal tesseral functions is 
d?y]r o . . 4m 3 —I 
dit 2 ^ 4 sinlAt 
xjj=0 
(7) 
There is one case in which the functions used above become nugatory—viz. : when a 
is zero, or the tores are such that R=r and they touch themselves at the origin. In 
this case the proper curves are the two orthogonal families of circles, touching, the one 
set the axis of z, and the other the axis of p at the origin—-viz. ; 
