Fundamental Law oj Electrical Action. 353 



which plays the same role in four-dimensional analysis as 

 the familiar operator V in three-dimensions 



was called by Minkowski " Lor," in honour of H. A. Lorentz, 

 the discoverer of the Principle of Relativity. 



It is denoted by □• 



The operator 



V3* s + ~df + d~~ 2 + dl 2 ) ' 



which corresponds to the three-dimensional operator 



is genera! iy denoted by Q 2 - 



The set of four quantities p(u l} u 2 , u 3 , i), where p = density 

 of electricity at a point, is a four-vector according to Lorentz 

 and Einstein. It is known as the Stream-four-vector and 

 will be denoted by S. 



4. 



The potential-four-vector a satisfies the equations * 



Q*a=— 4tts, or n 2 a = 0, . . . (1) 



according as the world-point at which n 2 a is taken is 

 occupied by a stream-four-vector or is empty, 

 a satisfies also the equation 



Div. a, or fna) = (2) 



at all points of the world-space. 



Now the fundamental solution of equations (1), due to a 

 single stream-four-vector S, occupying the world-point 

 (V, y\ z\ V) is 



As a ° to\ 



01' A" fNO , , rv-o— -, >w . /7 TF7o, {<>) 



[ x ~a:y+{y-y'y + {z-zy+(l-V? 



where (#, y, z, I) is the world-point at which a is to be 

 estimated. 1 



A can be proved to be equivalent to 



IT 



* Born, Ann. d. Physik, vol. xxviii. p. 571. 

 Phil. Mag. S. 6. Vol. ?>1 . No. 220. April 1919. 2 C 



