192 



Prof. A. Anderson on the Mutual 



This must be integrated over the whole half plane on one 

 side of the base AB. 



There aro two integrals to be found, 



f 



cos u sin "a 



P 



d& 



i sin *cc Q 

 ma 1 . do. 



J c 



They present no difficulty, the first being obtained by re" 

 ducing to polar coordinates, A being pole and AP radiu s 

 vector. The second is very easily got by taking as element 

 dS the difference between the areas of two segments of 

 circles on AB, one containing an angle « and the other an 

 angle ct + du. Their values are respectively 4c and 2c. The 

 total mutual energy is, therefore, 



Consider now the case of two charges, e 1 and e 2 , at A and 



Fis:. 2. 



B V 2 



B both moving in the direction AB (fig. 2) with velocities 

 jvi and v 2 . The mutual energy per unit volume at P is 



fie&tws. 



4-7T 



sin $i sin 2 jr^r 2 2 . 



And the mutual energy in the ring traced out by dS at P is 

 d8 .t^-p^ , p sin 6 1 sin 0,/fiV 



~ ' 2 rx'ra 3 ' 2 c 3 



But fsinWS = 2c 2 . 



Hence the total mutual energy = a 2 , that is, double 



what it is in the former aase for the same values of charges 

 and velocities. 



