132 Mr. G. F. C. Searle on the 



Thus since, by (13), U + T, the total energy of the sphere 

 in steady motion, is W-f-U , we have 



u+T= ^(£l og ^_l\ = ^(i'lo g ^-i). 



tci \u- b v — u J 2Kft\u °v— u J 



This value is identical with that which I found* by inte- 

 gration throughout the whole field of the sphere in steady 

 motion. 



Since the sphere moves along an axis of symmetry, the 

 momentum carried off in the pulse is, by (38), 



p_ pun Q 2 P^ sin 3 6 cos 6 dO 

 1(6 J ( 1 — n cos 0) 2 



Putting 1 — ncos = 7i and using fiKv 2 = l, we find 



X T lb 



p_ Q 2 fjV-1 3-n 2 „ ,1 



1— n 



= q?» {(3-".:)ioo" +tt -^'v 



4att?& LV « / v — u vj ' 

 " S3?\3 . 5 + 5 . 7v 2 + 7 ."9tf* -/ 



<Z/<, 



Hence, as was proved in § 18, for small values of u/v the 

 momentum carried off by the pulse varies as w 3 . 



From the values of W and P those of U, T, and M can be 

 deduced by the method explained in § 15. 



§ 20. Sphere ivith a uniform volume-charge. — If z be 

 measured from the centre of the sphere, we have, in this case, 



dQ 3Q(a 2 -s 2 ) 



Hence f + VQ\ 2 7 3 Q 2 



dz 4a 3 



and thus, by (37), 



w _ fiQ 2 u 2 6 ft ^ sin 3 # </# 



4a ' 5 Jo (I — '/i cos 0) 2 ' 

 and hence, by § 19, 



TTT 6 av 2 Q 2 /v, v+u \ 



W=- . — ; — -log 2). 



2a \u v — u J 



* Proc. Royal Soc. vol. lix. p. 344 (1896), and Phil. Mag. Oct. 1897, 

 p. 340. 



