without the Equivalence Hypothesis. 113 



to say, including pure space-rotations and shifts of origin, 

 the Lorentz group is a ten-parametric one, exactly as the 

 above group. It remains only to show that the latter does in 

 fact include free shifts of the origin of the four coordinates 

 x u a' 2 , #3, # 4 (the fifth being only an artifice for simplifying 

 the investigation) . 



Now, return to (19 a) which, written out fully, are 



a?i , =« u # 1 +a 13 #2+ +ai 5 #5^| 



' / " ' ' ' y. . . (19a) 



#1 =a 4 l#l + ff42#2+' +«45«5 | 



IBs! =%i#i + a 62 # 2 + + a 55 a; 5 J 



The origin of the ^-system is x x ^=x 2 =#3= # 4 =0, and by 

 (17), x 5 2 = R 2 , say # 5 = -f;jft = V + l, according as the world 

 should be elliptic or hyperbolic. Thus the origin 0' of the 

 ^'-system will be 



ai=ai 6 Ii, x 2 ' = a 25 R, etc., x 5 ' = a 55 R, 



satisfying (17) in virtue of the fifth of (21). Thus by 

 ascribing appropriate values to a 15 , . . . . a 45 any world-point 

 can be made the origin of the new system. Q.E. D. 



An interesting feature is that (19 a) with (21), although 

 having the outward form of an ordinary rotation with 

 " fixed origin/' yet contain also shifts of the world in itself, 

 unlike the Lorentz transformations if written, in four vari- 

 ables, ^i=a lK x K . The simple reason is that our formulae 

 do express a rotation round a fixed point ; a point not of 

 the world, however, but an extraneous one, in the fifth 

 dimension, or, to speak figuratively, "inside" the sphere 

 whose surface represents the world. The contrast with the 

 Lorentz six-parametric ' : rotation " can perhaps be best 

 illustrated by comparing an ordinary plane with an ordinary 

 spherical surface. And the analogy fits because the Min- 

 kowskian world is flat, while that which concerns us here 

 is assumed to have some constant curvature. 



Having thus ascertained the properties of the full, ten- 

 parametric, group of natural transformations in their simple, 

 " kinematical " form, it will be enough to develop the 

 analytical expression for the sub-group only, corresponding 

 to a fixed origin of a? 1? a 2 , #3, # 4 . To obtain (7 = 0, write, 

 in (19 a), 



ai5 = «25==«35 = a 46 = 0, a 55 = l. 



Phil Mag. S. 6. Vol. 36. No. 211. July 1918. I 



