240 Mr. J. J. Thomson on the Electric and Magnetic 



The part of the kinetic energy we are concerned with will 

 evidently be 



+ \f ) dz r dx)li\dz dx) 

 ( d d\l/dG' dF'\-\ 



+ (*s -* *;) hU» - *r)J dx A y dz 



Let us take the first integral first, and take the term depend- 

 ing on p ; this is 



4tt J J J dz R V dz dx } dy It \ di- dy J J 



Integrating by parts this becomes 



-if JJ F (i i dydz+ i n d * d * + ii***) 

 + ^fli (-£' * rf * + S dx dz + % dz **) 



fiepCCC 1 d (d¥ r d& , dW \ 

 + 4v}jJR.dtB\dtc + dy + dz ) 



- W {i^ + if + d%)n dxc, y dz - 



The surface-integrals are to be taken over the surface of the 



sphere; and the triple integral is to be taken throughout all 



space exterior to the sphere. 



If the sphere be so small that we may substitute for the 



dW 

 values of F / , —, — , &c. at the surface their values at the centre 

 dx 



of the sphere, the first surface-integral =fiepF\, where ¥\ is 

 the value of F' at the centre of the sphere ; the second surface- 

 integral vanishes, and the triple integral also vanishes, since 



dx 2 ~EL + dy 2 H + dz 2 B > 

 and 



dW d(¥ dW =0 



dx dy dz ~~ 



