131 



Mr. G.F. C. Se&vleonthe 



times the values for the outer sphere alone, and hence, by 

 § 15, the values of U, T, and M are (a — b)jb times the values 

 for the outer sphere alone. 



In this calculation a uniform distribution of the charges 

 has been assumed. But this is not the natural distribution, 

 corresponding to the speed u, for a pair of conducting spheres. 

 So far as I know, the distribution in the latter case has not 

 been investigated. 



§ 22. Sphere toitJi surface-density proportional to cos 6. — 

 Let er, the surface- density, be given by cr = 0- o cos#. Then 

 the same distribution can be obtained by taking two spheres 

 of volume-densities p and — p with their centres at a distance 

 f apart, and proceeding to the limit | = 0, while pj has the 

 constant value cr . In terms of g, the "electric moment" of 

 the sphere, we have 



P?=0"o = 



■kird z 



Let (fig. 3) be the centre of the sphere of density p and 

 0' that of the sphere of density —p. Then, if A be the area 



Fio-. 



of the section of the sphere by a 

 plane at a distance z from 0, the 

 normal making an angle 6 with OX, 

 we have A=ir(a 2 — z 2 ). The area 

 of the section of the sphere 0' by the 

 same plane is ir{a 2 — (2 + fcos #) 2 }. 

 Thus the charge dQ contained be- 

 tween two planes at a distance dz 

 apart is 



dQ=7rp{a 2 -z 2 \dz--7rp{a 2 -(z + £ cos OyldZj 

 whence, to the first power of f, we have 



dQ/dz=2irzcoa0.p(;. 



Hence, since pi; = 3^/47ra 3 , 



dq 



dz 



?)</z cos 6 

 taF~' 



Since 



we have, by (37), 



W: 



z 2 dz— ^a z , 



ZfiuY C* si n* cos 2 Odd 

 S" Jo (l-^cos0f 



