130 Mr. G. F. C. Searle on tie 



§ 23. Ellipsoid of revolution with a surface-charge. — When 

 a charged ellipsoid is in uniform motion, the charge is dis- 

 tributed in the same way as when it is at rest*. The surface- 

 density is everywhere proportional to the thickness of an 

 infinitely thin ellipsoidal shell bounded by similar ellipsoids. 

 the inner surface of the shell coinciding with the ellipsoid. 

 Such a shell can be formed by uniformly straining a shell 

 bounded by concentric spheres. If the spherical shell be cut 

 by a series of parallel equidistant plnnes, they will divide the 

 shell into portions of equal volume, and hence the series of 

 parallel equidistant planes, into which the first series is trans- 

 formed by the strain, divides the ellipsoidal shell into portions 

 of equal volume. Thus, the charge contained between a pair 

 of parallel planes which cut the p. , 



ellipsoid is, for any given direc- 

 tion of the planes, proportional 

 to the distance between them. 

 Hence, if the distance between a 

 pair of parallel tangent planes 

 AA', BB ; (fig. 4) be t >, the value 

 of dQ/dz, where dz is an element 

 of the normal to the>e planes, is constant for those planes and 

 is equal to Q/jt? 3 where Q is the total charge on the ellipsoid. 



§ 21. For simplicity, the ellipsoid of revolution will be 

 supposed to move along OX, its axis of revolution. If «, h. 

 h be the axes of the ellipsoid, then, when the normal to the 

 tangent planes makes an angle 6 with OX 



p- = ±{b- 2 +(a 2 -b 2 )co* 2 0}. 



Since dQ/dz is independent of z, we can integrate (37) at 

 once with respect to ~, and thus, writing cos = zc, we find 

 for the energy carried off in the pulse 



/x?rQ 2 f (1 — x*)dx 



4 J-i (I-n*)V* + (^-AV 



== ^Q 2 { _J 1 + 2 J 2 -(1-^)J 3 } ? 



e ±7i 



where 



_ r +l _ <i,r _ j = f +1 dx 



)_, (1-^.r) 2 \'P+(a*-P)&' 



aM J,: 



* W. B. Morton, Proc. Physical Society, vol. xiv. p. 180 (1896), or 

 Searle, Phil. Mag. Oct. 1897, p. 331. 



