8 Mr Basset, On the Potentials of the surfaces formed by [Oct. 25, 
Let wh = yp, then 
v= xe Py, By le x4 
da* "dp" “ pdp, p* 
from which it follows that y sin @ is a potential function*. Now 
at the surface of the solid 
b= 2Vp%, 
. ysin 0=4Vpsin 0. 
Hence ysin@ is the potential of the induced charge, when the 
solid is placed in a field of force whose potential is 
—3Vpsin 8, 
and vr = yp = Up cosec 8. 
Hence from (9) 
yo ens BoM P20 Pw, 
Parr ie 
9. Since a parabola is a limiting form of an ellipse, and con- 
sequently a cardioid is a limiting form of an elliptic limagon, it 
follows that the results of the preceding portion of this paper may 
be modified, so as to give the corresponding results in the case of 
paraboloids and cardioids of revolution. 
In order to see how the transition takes place, from a prolate 
spheroid to a paraboloid, let us consider the case of a symmetrical 
potential. 
The function Q, (v) ee the equation, 
Se) 0 (11). 
* The complete integral of (10, a) may be expressed in the form 
vie= |" cos ef (z+ up cose) acs [* cosh ef (2+ 1p cosh e) de. 
0 
For transforming to polar co-ordinates, (r, cosy), it is easily seen that a 
solution of (10, 0) is 
x= {40 PI (u)+ — Qh Ww} . 
(n+1) 
Now P,= zo" {utr/we—1 1 cose}" cos ede, 
oral gi= Ap cosh ede 
o (#+,/u2—1 cosh e)"+4 
whence the result at once follows. 
