Dynamics of the Electron. 81 



Therefore we have 



X z = g^ [ (a 2 w 3 — a 3 w 2 ) 2 + (« 3 ^4 — « 4 ^s) 2 + (a 4 ^ 2 — «2^ 4 ) 2 



— (ajWg — « 2 ^i) 2 — («iW 3 — «3^l) 2 — O1W4 — ^4^l) 2 ] ' 

 NOW putting a 2 _ ^2 + a2 2 + ag 2 + ^2 



and using the identity 



a^i + oi 2 w 2 + ol z w z + a i w i = 0, 

 we easily prove that 



X,= |^[-« 2 (l + 2V) + 2«i 2 ]. 

 Similarly 



e 2 

 X v =j-[—W 1 W 2 ot 2 + oi 1 oi2], &c. 



We shall now calculate the total force on the space exterior 

 to the electron. According to the Principle of Relativity, 

 this space must be uniquely defined. In our case, this space 

 is perpendicular to the axis of motion of the electron, and is 

 hounded on the inside by the surface of the electron. The 

 external boundary is at an infinite distance. Let d£l denote 

 an element of volume of this space. Then the total force is 

 given by 



Now since a, and consequently / 12 ,/ 23 .../ 4 ..., are functions 

 of the relative distance [(#— #'), (y— y'), (z— z'), (I — I')], 

 we have 



da?' ~ d# ' 



Therefore 



f *= - ftf x ^ + !Jv°+ &fc»+ ?if**\l • 



PAtV. 1%. S. 6. Vol. 36. No. 211. July 1918. G 



