General Relativity. 223 



quadratic form from which the gravitational equations are 

 derived is the same as the quadratic form which determines 

 the optical properties of the medium. Indeed, the example 

 which I considered on p. 262 of my first paper would seem 

 to indicate that this was not the case. It should be mentioned 

 that in the first seven equations in this example there is a 

 misprint, e/x should be replaced by (e/x)" 1 . On the above 

 view Einstein's idea of an influence of gravitation on light is 

 simply an hypothesis, but a very interesting and reasonable 

 one. It may be remarked, however, that in the theory of 

 surfaces there are two fundamental quadratic forms, and we 

 may perhaps expect something similar in general relativity. 



With regard to possible extensions of the idea of relativity 

 it may be worth while to consider transformations analogous 

 to the contact transformations of dynamics in which the 

 co-ordinates x, y, z, t and the component velocities u, v, w 

 correspond to a new set (xiyiZjt^v^) in such a way that 

 the differential equations 



dx l _dy 1 _ eki _ dt 

 U^ Vi w x 



are a consequence of the equations 



dm _ dy _ dz 



U V IV 



dt. 



This may be secured by making a single quadratic form, 

 such as 



(dx 2 + dy 2 -+• dz 2 - c 2 dt 2 ) (c 2 ~ u 2 -v 2 - iv 2 ) 



+ (c 2 dt — udx — vdy — wdz) 2 , (c 2 >u 2 + v 2 + w 2 ), 



an invariant*. Various other quadratic forms consisting of 

 sums of squares maj, of course, be adopted instead. 



H. Bateman. 



Throop College. 

 Pasandena, Cal. 

 Aug. 10th, 1918. 



* This is a positive definite quadratic form in the variables dx — ndt, 

 dy—vdt, dz — wdt, and so can only vanish when all these quantities are 

 zero. 



