WEBSTER. — PROPERTIES OF THE ETHER. ' 519 



(25) 

 j { (P. Vx Vx5P — 6'P. VxVx5P) + (E.5E — C;E.5E + 8U)}dTdt = 0, 



t, 



or, since 8U = and 



(26) VxVx5P = — V-5P = +5E, 



when no motions of charges have been varied, 



(27) I J yj HP + E).5E - G'(P + E)-5E} drdt = 0. 



<, 00 



+ 



SpHtting E into the parts E^ and El, treating E hkewise, and applying 

 Green's Theorem as in (14), 



(28) j j j j \{V+ Es)'bEs - 6'(P - is)'Sks]dTkt = 0, 



+ 

 because 5Ez, and 5El are zero when no charge motions are varied. 



(29) Therefore P + E^ = 0, 



(30) and P + Es = 0, 



(31) or 4= -y Pot(E + pP). 



47r 



(32) and E^ = — ^ Pot(E + p p). 



47r 



Applying Vx to (29) and (30) we have 



(7) VxE = — H, (8) VxE = — H. 



To derive equations (9) and (10) from equation (11) or (12) let us 

 suppose that, for a short time during the interval ti and t2, an infini- 

 tesimal positive charge dc, occupying a small tube of length dx' and 



+ 

 cross section da- ^ and lying in the direction of p is displaced in some 



other direction through a distance 5r. To satisf}' equation (1) with 



+ 

 this variation we may superpose on the actual value of E a straight 



5 Any cloment of surface may be considorod as a vector along its normal, 

 and when its direction is chosen, the jjositive direction around its boundary is 

 that of a right-handed screw rotation. 



