198 Mr. G. H. Livens on the Principle of 



<f> and A to satisfy the restrictions which necessitated their 

 introduction. The complete variation is easily effected and 

 is, in fact, 



J> [{£(8) -£-£♦;(■#)}" 



+ { }% + { }*g+!(AJM+A&+AJt)] 



+ ■— C 2 dt i dv [(BSB) - (ESE) + 4> div SE 



-(ACu r iaB) + i(A^)]- 



On removal by integration by parts of the various differ- 

 ential operations affecting the independent variations and 

 noticing that the time differentiations affecting the coordinates 

 of an electron are total differentiations following the motion 

 of that electron, we get finally for the total variation 



+lf}*'+( ><*+ ] 



+ hS'* dt i dv -^ B " Curl A ' 8B) ~ ( E +grad * + lifi'' SB )1 



- T- f h dt Qs {</>SE, - [A8B] B } 

 47r J'i J 



+ I X - (AA* + A y fy 4- A,8s) - X (A SE) I *. 



I € C \ ti 



The integral with respect to S is taken over the infinitely 

 extended surface bounding the field. 



The variations Ban, St/, 8z, which give the virtual displace- 

 ment of an electron e, and the variations SE, SB, which 

 specify the condition variations in the sether, can now be 

 considered as all independent and perfectly arbitrary ; hence 

 the coefficients of each must vanish separately in the 

 dynamical variational equation. We conclude that (p and A 

 must tend to zero regularly at infinity and then at each 

 point of the field 



E+-|4+g rad< £= > 

 c ot 



B-CurlA=0 ; 



