without the Equivalence Hypothesis. Ill 



among those required by (B). These equations, common to 

 (A) and (B), are 



a ii- + a L + • • • + a L~ *> five ec i s '' 



and 



in all 15 equations. In addition to these (B) itself gives 

 the condition 



«l 2 + a 2 2 0+----+ a o 2 0= 5 



and five more equations of the type 



a io«i K + «20 a ac + ---- + a 5o«5K=°- ■ • ( 20 ) 



The latter, however, being a system of homogeneous equa- 

 tions for the a l0 , we have either det. (20) = 0, when (20) are 

 reduced to four independent equations only, or det. (20) =£0, 

 and «io = ^2o— • • • • =#5o = 0* I' 1 t ,ne former case we have in 

 all 15 + 1 + 1 = 20 equations for 5x6 = 30 coefficients, and 

 in the latter case, the first 15 equations only for 25 coeffi- 

 cients. Thus, in either case the coefficients can be expressed 

 by 10 free parameters, or the transformations in question 

 are ten-parametric. Without sacrifice of generality we can 

 take the second case, i. e. a t0 = 0, and therefore, the homo- 

 geneous transformations 



<=<V*v *=1,2,... 5, .... (19a) 



with 15 equations 



aJ + a.J+...-la.J=l, 



(21) 



for the 25 coefficients a u , a l2 , ■■■ a 55 . And, as every pair of 

 transformations (19 a) can be replaced by a single trans- 

 formation of the same kind, the said natural transformations 

 constitute a group, viz. a ten-parametric one. The relations 

 (21) are exactly of the same form as the six equations which 

 are well-known in connexion with the ordinary transforma- 

 tion of Cartesian coordinates by a rotation of the system, or 

 the 10 equations connected with the Lorentz transformation 

 (with fixed origin of time and space). In fact, (21) taken 

 by themselves would correspond to a typical orthogonal 

 transformation in five variables. Since, however, our five 

 coordinates are not independent but bound to one another 

 by (17), our case is better expressed by saying that it is the 

 four-dimensional analogy of the (rotation or) motion in 



