[ 296 p.. 



XXXV. Note on the Derivation from the Principle of Rela- 

 tivity of the Fifth Fundamental Equation of the Maxwell- 

 Lorentz Theory. By Richard 0. Tolman, Ph.D., In- 

 structor in Physical Chemistry at the University of Michigan* . 



IF we consider two systems of " space time coordinates " 

 S and S' in relative motion in the X direction with the 

 velocity v, any kinematic phenomenon which occurs may be 

 described in terms of the variables a, y, z and t belonging 

 to the system S or x\ y', z' and t' belonging to the system 

 S'. The Einstein theory of relativity has led to the following 

 equations for transforming the description of a kinematic 

 phenomenon from one set of coordinates to the other f. 



^vhwi'-y) • ■ ■ ■ « 



*= vfe^(*-*° (2) 



y'^y - • (3) 



*"=* W 



(where c is the velocity of light and fi is substituted for the 



fraction — ). 

 c 



The content of these equations may be expressed in words, 



by saying that an observer in the moving system S' (S having 



been arbitrarily taken as at rest) uses a metre stick which, 



although the same length as a stationary metre stick when 



held perpendicular to the line of relative motion of the two 



systems, is shortened in the ratio of V\— ft 2 : 1 when held 

 parallel to OX, that clocks in the moving system beat off 

 seconds which are longer than those of stationary clocks in 



the ratio 1 : \/L—/3 2 > and that a clock in the moving system 



which is a} units to the rear of the one at the centre of 



v 

 coordinates is set ahead by rf-% seconds, although the two 



c 



clocks appear synchronous to the moving 'observer. A simple 

 non-analytical derivation of these relations has been given in 

 another place J. 



Let us now take the Maxwell-Hertz equations for the 



* Communicated by the Author. 



t Einstein, Ann. d. Physik, xvii. p. 891 (1905) ; Jahrbuch der Radio- 

 aktivitdt, iv. p. 411 (1907). 



X Lewis and Tolman, Proc. Amer. Acad. xliw p. 711 (1909) ; PhiL 

 Mag-, xviii. p. 510 (1909). 



