Mathematics. — "On the Light Path in the General Theory of 

 Re/ativiti/." By Piof. W. van der Wouük. (Communicated 

 by Prof. H. A. Lorentz.) 



(Communicated at the meeting of September 30, 1922). 



In Einstein's tlieorj the path of a ray of light is found bj piiHiiig 

 the condition that it is a geodesic null line in the four-dimensional 

 time space ')• If accordingly we represent the line element of this 

 time space by 



^*' = ^ 9ik ^•^. 'H ^^) 



i,lc 



the light path satisfies equally the equations of the geodesic as 

 those of the null line 



ds = (2) 



As far as we know the remarkable relation existing between 

 these differential equations, has not yet been pointed out. We shall 

 prove that lliis may be expressed in the following way: 



a geodesic hewing one element, i.e. one point loith the tangent at that 

 point, in conwion with a null line, is itself a mill line. 



h\ order to prove this we shall fir-st give the equations of the 

 geodesic a form different from the usual one (§J), as on account 

 of (2) it is not desirable to take .s- for the independent variable. 

 With a view to an application which we shall give later on, we 

 take one of the coordinates of the time space for the independent 

 vaiiable. 



We shall conclude by pointing ont the (evident) physical meaning 

 of the theorem. 



^1. If the line element is represented by 



ds^ — 2 g.^ dx. dxj^ , 



i,k 



the equations of the geodesic are 



- ^ ^^j^-f^j^f-.^^ (3^ 



ds^ ;,// I I' 1 ds ds 



') From this there follows for the statical field (gik independent of the time- 

 coordinate Xq and goi = for I ^^ 0) the principle of Fermat for the minimum 

 time of light in three dimensional space. 



