General Relativity and Einstein's Theory. 393 



to the X L plane. Let vS" denote the system under investigation. 

 If the transformations between 6' and a EucHdean system S' with 

 constant hght speed are sought, (11) takes the form 



ds- = dx'^ + dl'^ = hHx^ + km-. ( 14) 



Without loss of generality, it is possible to put 



dx' = pdx -\- nqdl, 



dl' = — q dx -\- npdl. (15) 



Then 



h^ = p^-^ q\ 



k-" = n\p- + q-) . 



Making use of the fact that li and k are not functions of /, and 

 that dx' and dl' must be exact differentials, it follows that 



I ndx 



(16) 



I ndx 



^ — -^ „ o-U n ■ ^ 

 q = — e ^ sin , 



n a 



where A and a are constants, and the square of the linear element 

 becomes 



2 ndx 



ds'=^e ^J " [dx'^+n'dr). (17) 



These values of p and q define the only type of system for 

 which h and k are not functions of /, other than the group of 

 Euclidean systems reciprocal to S' which have already been dis- 

 cussed. It will be shown now that if a is positive the system 5" 

 defined by (15) and (16) is accelerated in the negative X 

 direction relative to S' . Consequently a point at rest in vS"', or in 

 a system reciprocal to S', appears to be accelerated in the positive 

 X direction when its motion is referred to the reference frame 

 of 6*. 



For 



and 



/dx\ q I 



(-77) ,= —'/- = — ''^ tan-, 

 \dl/x' p a 



d [dx\ n / dn ^ I 



3> I "T/ ) 1 = sec — h ^' r.- tan — • 



dl\dl/x a a dx a 



