1886.] Mr Basset, On the Potentials of Circular Discs. 425 



(4) On a method of finding the Potentials of Circidar Discs by 

 means of BesseVs Functions. By A. B. Basset, M.A. 



1. The present paper was commenced last summer, with the 

 object of developing a method for finding the potential of an elec- 

 trified circular disc by means of definite integrals involving Bessel's 

 functions, when the disc is placed in a field of force whose potential 

 is given. The same problem has been recently dealt with in a 

 similar manner by Mr Gallop in the Quarterly Journal*, but as my 

 own method is somewhat more general than his, it may be worth 

 while to lay this paper before the Society. 



Let us take the normal through the centre of the disc as the 

 axis of z, and employ cylindrical co-ordinates z, p and (p. The 

 potential of the field of force may be expanded in a series of terms 

 of the type 



f n (z, p) cos m<f> (1). 



If V be the potential of the induced charge, V will consist of a 

 series of which the term corresponding to (1) may be written 



V m = \F(X)e-^J m (\p)cosmcp (2). 



There is nothing to determine the value of X excepting that it 

 must be positive on the positive side of the disc, and we must 

 therefore suppose X to have all values from oo to 0, and replace the 

 sum by a definite integral, whence 



V m = cos m^> r e -^F (X) J m (Xp)dX (3). 



Jo 



At the surface of the disc f m (z, p) will be a given function of p 

 only, say — <f> (p), hence if c be the radius of the disc, we 

 must have 



/.CO 



cp(p)= F(X)J m (Xp)dX (4), 



J 



when p < c. 



The density which is proportional to — dV/dz must vanish 

 when p > c, hence we must have 



f> 00 



0=1 XF (X) J m (Xp) dX (5), 



when p > c ; and the solution of the problem consists in deter- 

 mining F (X) so as to satisfy (4) and (5) when $ (p) is given. 



I have only succeeded in obtaining the solution of this general 

 problem in the two cases in which m = or 1. The first case can 

 be deduced by means of the theorem of Art. 2 and a theorem of 

 Mr Gallop's ; the second case is dealt with in Art. 5. 



* Vol. xxi., p. 229. 



