Potentials due to Moving Electric Charges. 135 



LP^ intersect the plane xYy in a line which makes an angle 

 (f> with P^-. Denoting the angle OPM by a, we have 



r 1 2 + r 2 2 = 4& 2 cos 2 a + 2(a 2 -6 2 ) 



= 46 2 [ncos(9 + /sin(9cos(/) + ??isin(9sin0] 2 + 2(a 2 -/> 2 ). 



Hence 



+ a 2 - b 2 } sin 0d0d<j> 



A = p 1 1 {2& 2 (ftCos#-l-Zsin #cosc/> + msin 0sin0) 2 



*^*' / u 2 



or, if the co-ordinates of P in reference to the axes through 

 are ,v, y, z, 



^ra 2 -x 2 -y 2 -z 2 } sin 6 d6 d(j> 

 A = p( i {2(0cos^4-«^sin 0cos<£+.y sin #sin<£) 2 



l-^sin 2 



c 2 



the limits of being — - and =■ , and those of 0, and it. 

 Hence 



d 2 A a»A , B 2 A . f* r- sin0rf< 



v 1 -?™'* 



T 17 sin <9 rf<9 27rpc , c + ?/ 



= — lirp \ J = = — lo ~ 



In 



Vi-fi 



Q 



C — U 



sin 2 



If in the expression for A we make w = 0, ?/ = 0, z = 0, we 

 get for the value of A at 



7rpa 2 c , c + u 

 u °" c — it ' 



d 2 A 

 which vanishes with a, 2 , having the same linear 



dimensions as A, also vanishes with a. 

 It follows, therefore, that 





A 



2tt? 



