to the Motion of Electrification through a Dielectric. 331 



Here the numerator of the fraction to the right is the vector- 

 potential of the convection-current. Calling it Aq, we have 



A.=^=2^ m 



Inserting in (18) and expanding, we have 



A={l + (2J/vSjy + (p/v\/y+ ...}Ao. . . (20) 



Given then pu as a function of position and time, Aq is known 

 by (19), and (20) finds A, whilst (15) finds H. 



10. When the motion of the electrification is all in one 

 direction, say parallel to the x^-axis, u, Aq, and A are all parallel 

 to this axis, so that we need only consider their tensors. 

 When there is simply one charge 5- at a point, we have 



Aq = qu/r, 

 and (20) becomes 



A = q{i + {p/vS/y + (p/vS/y+ . . .}(w./r) . (21) 

 at distance r from q. When the motion is steady, and the 

 whole electromagnetic field is ultimately steady with respect 

 to the moving charge, we shall have, taking it as origin, 



p= —u(d/dz)= —uJ) 

 for brevity; so that 



A=qu{l + (wD/t;V)'+ (wD/vV)'+ • • .\r-K . (22) 

 Now the property 



VV+2 = (7z-|-2)(w + 3)r'' .... (23) 

 brings (22) to 



^=H'r*">^^P'ri^ ■■■}■'■ (^*) 



and the property 



D^^r^'^-i =12.32.52. ..(2n-l)V7r, . . (25) 



where v=sin^, 6 being the angle between r and the axis, 

 brings (24) to ♦ 



^=.{l+?2(l+i?i^(l+?6^'(l+---);M 

 which, by the Binomial Theorem, is the same as 



A=(^t//i'){l-MVy}-^, . , . . (27) 

 the required solution. 



11. To derive H, the tensor of the circular H, let n/ = A, the 

 distance from the axis. Then, by (15), 



-rj clK clK , iiv dK quv(^ , d\{^ u^ A-* ,^„, 



