1886.] Circular Discs by means of BesseVs Functions. 427 



3. If a charged conductor of the form which we are consider- 

 ing be placed in a field of force, the density will usually be 

 infinite at the edges, but dVjdz will always be finite except at the 

 edges ; whence although it is necessary that <p (p) should be finite 

 and continuous between the limits p and q, it is not in general 

 necessary that it should be finite at the limits. There are however 

 two special cases, viz. (i) q = 0, p finite ; and (ii) p — cc , q finite, 

 which require separate consideration. 



The first case is that of a circular disc of radius p ; and if 

 <p (p) became infinite when p — 0, there would be a singular point 

 at the origin. 



The second case is that of an infinite plane screen having a 

 circular aperture, and if (f> (p) became infinite when p = go , the 

 density would be infinite at an infinite distance from the aperture, 

 which seems to be physically impossible. 



If therefore in the first case <p (p) = go when q = 0; and in the 

 second case <p (p) = go when p = oo , the theorem could not be 

 safely employed. 



It must also be borne in mind that although the reduction of 

 the integral in the form given may not always be easy to effect, 

 yet as a matter of fact, the integral is really the limit of 



f "e-^Xucp (u) J m (Xu) J m (Xp) du, 



J q 



when z = 0, and we may therefore reduce this latter integral to 

 a simpler form, whenever it is possible to do so, and then put 

 z = 0. 



4. If $ (p) is finite and continuous for all values of p between 

 and cc inclusive, we may put p = x , q = 0, and the theorem 

 becomes 



<p (p) = I d\\ Xucp 0) J m (Xu) J m (Xp) du 

 Jo Jo 



for all positive values of p. In order to solve the problem of find- 

 ing the potential of an electrified circular disc which is placed in 

 a field of force, we must determine a potential function U m cos m<p 

 which satisfies the conditions that when z = 0, 



^m = Jm (V) P < C > 



■-j* = p>c, 



dz r 



and the potential of the disc will be 



tcosmcp! dxf U m Xu<p(u)J m (Xu)J m (Xp)du....(7). 



J I) Jo 



