MR, W. M. HICKS OK TOROIDAL FUNCTIONS. 
613 
Tlie first prodaces functions analogous to those discussed in this paper—the second 
spherical harmonics of argument ^ — v, and order The surfaces ?; = const, give 
confocal spheroids. Since \ / p = va cosh u cos v, it will result that </> is expressed as 
the sum of terms of the form {AP(m)+BQ(m)} (CP'^ + DQ^r)} cos (mw+/3), where 
T v , Q' are spherical harmonics of argument ^ — v, and P, Q are spherical harmonics of 
imaginary argument. 
In the applications that follow it happens that u, v are such that p~£ l is a function 
of u only, say f(u); in this case we obtain solutions by putting = cos 
where 
Cp-^r / 0 4m 3 — 1 \ 
du°~ V 1 4/(«) ) 
xP=0 
( 3 ) 
The solutions of this equation for m = 0 may be called zonal functions, for n— 0 
sectorial functions, and for m.n general, tesseral functions. If U,„.„, U/.« are two 
independent solutions of this equation the general value of (p is given by 
\ p(f)=’£X{AJJ„ lM cos ( nv-\-a ) cos A'U' mM cos ( nv-\-a ) cos 
If (f> be given over any two surfaces u= const., it is clear that the constants can be 
determined in the above by means of Fourier’s theorem. This will be more fully 
discussed in the sequel. 
2. Before passing on to particular cases, there is one remarkable result to be noticed. 
If in the equation transformed as above, we put \jj = \p' cos then t// satisfies 
the equation 
bvr 
+ 
tfjr' 
Tv j: 
Hence if xp' be any two-dimensional potential function, then i// cos (\w-\-y) is a 
three-dimensional potential function. Since this expression changes sign when id 
increases by 277 it is not suitable for all space; but a diaphragm must be supposed to 
extend from the axis of 2 to infinity in one direction, and to be impassable. Though 
the result is interesting it does not seem to carry important consequences, as there is 
not sufficient generality in the expression. We may choose the form of the surface, 
and certain other conditions, but all the surface conditions are not arbitrary. Thus let 
us take an anchor ring divided by a plane through its axis. Let us keep the curved 
surface and one end at zero temperature, then the distribution of temperature at the 
other end is determinate though its absolute magnitude is arbitrary. To prove this, 
we notice that if (a, h) be the radii of the circular axis, and generating circle respec¬ 
tively, and r the distance of any point from the circular axis 
