368 Mr. Megh Nad Saha on the 



But here we have 



~dx 



CIT 



t. £. in place of the time t, we have to use the proper-time r, 

 where dr = ^/(l — u 2 ) dt. 



The name proper-time for the function t, suggests that it 

 has some special relation with the time-coordinate, whereas 

 in fact it is perfectly symmetrical, and similarly related to 

 each of the four coordinates. To dispel any such false 

 notion, it is now usual to designate dr as an element of length 

 of the world-line of motion. Thus 



dr = ds = vdt 2 — dx 2 



■ df-dz 2 , 

 -1 times the 



direction 



and (ti^, i0 a , 101, w 4 ) becomes \/ ■ 

 cosines of the element ds. 



In a system consisting of two electrons only, the forces 

 controlling the motion are due to electronic attraction only ; 

 the gravitational field, being 10 -42 times smaller than the 

 electronic field, can be entirely neglected. Following 

 Minkowski *, the equations of motion can be written in the 

 forms : 



d*x _ d<£ d /_d<£ \ 1 , d 2 a _ "dp d /'b$\\ 

 cHl «ds 2 -lbx~ds\^clx) 

 ds 



2 d 2 y _ d<£ _ d_ / d<K 



cm °^"d.y ds \% d JL) 



ds 



m 



ds' 



'da ds' I ^ da J 



,d 2 b 30' d ,30' 



d6 ds' 



d\ 



€m °d? = ^ 



~dcf> ^/30 

 ds 



(*p 



(#) 



as 



d'H _^4> d r M 



dl 



c m °d? ~ N ds 



ds J 



ds 2 ~ ^c ds' Ldc) 

 ds 1 



~d\ ds' \^d\) 



K16) 



m —r~r = 



rf 



3 



<h* ; 



These equations are a particular case of the general 



* Minkowski, loc. cit. Appendix. 



