from an Electric Source. 



167 



surrounding medium. Of these functions only A will be 

 needed here. It will be given presently. 

 To abbreviate, write 



p = — > g(p) = sin pIp - cos P- 



(7) 



Then the mean values of ^KP 2 , ^KR 2 , ^M 2 , taken over a 

 period of oscillation, will be 



A 2 sin 2 6 



n 2 (psmp + g)\ 



4A 2 cos 2 



n'g* 



A 2 sin 2 





respectively. Integrate each of these expressions through 

 the volume of the sphere, add up, and write 



na/v — u. 



Then the mean energy of the source will be 



U 



where 



'smp+gy+2g* 



8tt AV 



(8) 



<*">-tt ( - 



+r 



}dp. 



(9') 



Develop the integrand and integrate by parts the terms 

 containing sin 2 p//s 4 and sin 2p/p z . Then the result will be 



~. . . 2cos2w — 3 , ocvo v sin2ii s'm 2 u , 

 G(u) - *> + + 2Si(2w) -t- -^ ^3-, (9) 



where Si is the so-called sine-integral, 



. 8i ( *)=r^*. 



Jo ll 

 For moderate values of x, even up to x = 17, 



SiG^^.r-Jfy + j'!,-...; 



numerical tables, from .r = up to ,r=10 7 , and a graph, have 

 been given many years ago by Grlaisher *. The limiting value 

 is Si(oo) =7r/2. % "But for the present we shall require the 

 values of the sine-integral for small or moderate values of 

 x only. 



Developing the several terms in (9) into series, the reader 



* J. W. Glaisher, Phil. Trans, clx. (1870), pp. 367-387 ; reproduced 

 in E. Jahnke & F. Emde's Funktionentafeln, Teubner (1909), pp. 20-21. 



