L or entz- Einstein Transformation. 351 



AT n . , . „ d 2 x d 2 if d 2 z d 2 u . , 



JNow the vanishing ot -r-j, -j~, — ^, -^ is the necessary 



u«S rt5 uS <ft$ 



and sufficient condition that \ds, taken along a p;ith between 

 two fixed points in four-dimensional space, shall not be 

 altered by small variations in the path. Thus we have the 

 proposition that if \ds is unaltered by small variations in the 

 path, ids' also is unaltered. But ds' = kds; therefore when 

 ^ds is stationary, \kds is stationary. Suppose now, if 

 possible, that k is not constant. Take a path of integration 

 such that ^ds is stationary, and k is not the same for all 

 points near the path as for neighbouring points on it. If then 

 the path be varied slightly, the ends being kept fixed, but in 

 such a way that the new path is always on the side where k 

 is greater, \ds will be unchanged to the first order, and {kds 

 will be increased. Hence the hypothesis is violated, and k 

 must therefore be a constant. To pass from this result to 

 k=l requires merely a permissible change of units. There- 

 fore we can assert that ds and ds' are equal. 

 Next, 



dx' _ d#' das "dy' dy "ftx 1 dz ~dx' du 

 ds' 'dx ds ~dy ds "dz ds ~du ds ' 



-r. \ , dx dy dz , du dx' . 



i3ut when — , -~. -— , and -=- are any constants, -r-r is 

 ds ds ds ds J ' ds 



t i. u ,c\ xj ~& x ' ^ x ' ~^ x ' ^ x ' 11 



a constant by (b). Mence ^r— , ^— , ^ — , ^— are all con- 



J v J ox qy Q2 ou 



stunts, and x is a linear function of x, y, z, and u. 

 Similarly y' , z\ and u! are linear functions *. Thus the 

 postulates (3) and (4) are proved, and the Lorentz-Einstein 

 transformation can be obtained from them by the method 

 given by Cunningham f . 



* It may be noted that this cannot be proved from (6) alone without 

 (2). For (6) merely asserts that every straight line in the S-hyperspace 

 transforms it into a straight line in the S'-hyperspace ; and this is 

 satisfied by any transformation of the form 



X f' y -f'--f' U - f 



where p v p-z, P3, p± are any linear functions of x, y, z, and it, and f ruav 

 be any function of x, y, z, and u whatever. 

 t ' The Principle of Relativity/ p. 49 (1914). 



2 B2 



