1886.] the revolution of Limacons and Cardioids about their axes. 3 
origin being that focus which lies on the negative or left-hand 
side of the centre. Also, if we put 
v=coshyn, mw=cos&, 
it is known that the potential V, of any distribution of electricity 
upon a spheroid, can be expressed by means of the following series, 
V1zZ.: 
(ee, is me Bea anges) 
at an external point; and 
n (Y) Dm) 
V=2>24, 5" sin (m0 +a 
=> ”% (y ) Es (“) ( ste i) 
at an internal point; where y is the value of v at the surface, 
and the functions P,” and Q,” are what Todhunter, translating 
Heine, calls associated functions of the first and second kinds 
respectively. 
The functions of the second kind can be expressed either in a 
series of powers of 1/y, or by means of definite integrals; see 
Heine, Handbuch der Kugelfunctionen, Vol. 1. ch. iv.; Messenger 
of Mathematics, Vol xn pata 
If we invert with respect to the origin, the spheroid will invert 
into the surface formed by the revolution about. its axis, of an 
elliptic limagon ; whence it follows that the potential at all points 
outside this latter surface will be of the form 
=" 334,° Pp Bats a P." (uw) sin (m0 +4,,)....+. (Dy, 
and at an internal point 
a 2 Sl Q, () pm sin (MO -+a_)...... oN 
r n Oo” (ry) (u ) ( >) ( ) 
where yw and » are respectively equal to cos € and cosh 7 as before ; 
but 
Die ipi= 2esec & (E=p bn) ass ee tees (3) 
where 2c is the constant of inversion. 
3. The foregoing results can also be obtained without recourse 
to the method of inversion, by means of Neumann’s transformation 
of Laplace’s equation. 
It therefore follows, that the solution of any problem, which 
consists in finding the value of a potential function, which has a 
given value at all points of the surface under consideration, is 
reduced by expanding a given function of w and @, in a series of 
1—2 
~_ 
