354 Mr. Megh Nad Saha on the 



Therefore the potential-four-vector at a world-point 

 (#, y, z, I) due to a distribution in the world-space of the 

 stream-four- vector S is 



„_ lffl'f sdx'dy'dz'dV 



HrJJJJ (•-•')*+ (y-/j' + (i-z'J' + G-J') 1 " l ' 



N.B. In modern methods of treating problems on Electro- 

 dynamics, the usual practice is to choose a unit of current 

 which is ^ \ir timos smaller than the ordinary unit, thereby 

 instead of having D 2 a=— 47rs, we have rj 2 a=— S. I have 

 stuck to the older method, because this is more convenieut 

 for our purpose. 1 



The fundamental solution — 9 seems to have been first 



i 



obtained by Poincare *. It corresponds to the solution - 



in three-dimensional problems on Potential, and is a parti- 

 cular case of the following general result first obtained by 

 Poincare t. 



If (#!, x 2 , ... #») be the coordinates of a point in space of 

 n-dimensions, the fundamental solution of the generalized 

 Laplacian 



(V_ v v\ Y=0 



is A 



..j, where 7» 2 = (^-.V) 2 + (^-^) 2 + ■•■(•^-^') 2 . W 



5. Potential-four-vector at an external point due to the 

 motion of a point-charge. 



By a point-charge is meant a charge having no extension 

 in ordinary space. In four-dimensions, however, it has 

 extension in one direction, viz. in the direction of the time- 

 axis if the electron be stationary, or along an axis making 

 an angle of (tan -1 w) with the time-axis, if u be its velocity 

 of motion. 



Let (#, y, z, I) be the coordinates of the point-charge, 

 which we suppose to have started from the origin at time 

 £ = 0. Then we have (#, y, z) = — \/ — l(u,, u 2 , u 3 ). Let 

 (a, 6, c, X) be the coordinates of the external point at which 

 the potential a is sought. According to the general theorem 



* Sommprfeld, Ann. d. Physik, vol. xxxiii. p. 663. 

 t TMorie du Potential Newtonien. 



