Effects produced by the Motion of Electrified Bodies. 243 

 The kinetic energy 



%y \J 4/ 



Now 



10 m 



rfy r l ™ dx dz r)\ Q 2 7 ' " 3 9' 



_ /*[ / d 2 1 d 2 1 , f/ 2 IN/Sr 2 « 2 \ 



,/,*! .(PI , d 2 1 W5/ 2 a /2 \ gOO^l 

 + e r rf.rV <fo?<W +W dxdzr'K 6 2 J + 3 /_]> 



with similar expressions for Gr and H. 



dt 4«7T 



Z 2 1 d 2 1 d 2 



e W ^T2 7, +^^rx,-+^ 



3 



dor r t&e dy r dx dz r 



^ ,( fd 2 1 , cP 1 L , d 2 IV] 

 V dx 1 r ax dy r dx dz r J A 



with similar expressions for -j- and -j-. Since the particles 



are supposed to be very small, we shall neglect those terms in 

 F which depend on a 2 and a! 2 . 



The part of the kinetic energy we are concerned with in- 

 volves the product e e' : let us first calculate that part of it 

 arising from the product of that part of F due to e with that 



part of y- due to e f . We shall take the line joining the 



particle as the axis of x; and for brevity we shall denote 



The coefficient of u u' in the part of the kinetic energy we 

 are considering 



= * JIf GS I + £) '" & 7 dx dy dz - 



Now, for values of r > K, 



r' ~ r dx r 2 ! dx 2 r ' ' ' ' 



Now, since 



dx 2 r'~ dx 2 r dx'r + "" 



dx n r K } r»+ l ** 



where Q„ is a zonal harmonic of the nth. order ; and since the 

 product of two harmonics of different degrees integrated over 



