2 Mr Basset, On the Potentials of the surfaces formed by [Oct. 25, 
The following communications were made to the Society. 
(1) On the Potentials of the surfaces formed by the revolution 
of Limacons and Cardioids about their axes. By A. B. BAsseT, M.A. 
1. The first part of this paper deals with the potentials of 
the surfaces formed by the revolution about their axes of the 
curves which are inverses of ellipses with respect to a focus, and 
which for shortness I shall call elliptic limagons, to distinguish 
them from the orthogonal system of curves which are the inverses 
with respect to the same focus of the system of confocal hyper- 
bolas, and which may be called hyperbolic limacons. 
The potential of any distribution of electricity upon the sur- 
face of a prolate spheroid can be expressed, as is well known, by 
means of a series of spheroidal harmonics; hence the potential of 
the surface formed by the revolution about its axis of an elliptic 
limacon can be expressed in a similar manner by means of the 
method of inversion. The same result can also be obtained in- 
dependently by employing a transformation of Laplace’s Equation 
due to C. Neumann*, which is reproduced in the 19th volume of 
the Quarterly Journal, pages 349 and 350. 
The potential of a paraboloid of revolution, and thence by 
inversion that of the surface formed by the revolution of a 
cardioid about its axis, which is dealt with in the second part, can 
be obtained either by regarding a parabola as a limiting form of 
an ellipse, or independently ; and it will be found that the result 
assumes the form of a definite integral involving Bessel’s Functions. 
The fact that the potentials of paraboloids and cardioids of 
revolution can be expressed by means of Bessel’s Functions, ap- 
pears to have been noticed by F. G. Mehler of Danzig, and Carl 
Baer (see Heine, Handbuch der Kugelfunctionen, Vol. 11. pp. 174 
and 292), but I have not as yet been able to obtain any of their 
papers. 
Part I. 
2. Choosing the axis of w as the axis of revolution, let a, p, 0 
be cylindrical co-ordinates of a poimt, and let € and 7 be con- 
jugate functions of « and p such that 
x — tp = 2c cos’$(E+ um), 
then it is known that the curves 7 =const., & =const. are a family 
of confocal prolate spheroids, and hyperboloids of two sheets; the 
* Theorie der Elektricitiits- und Wéarme-Vertheilung in einem Ringe. Halle, 
1864 
