118 Dr. L. Silberstein on General B.elatimty 



geometry *. According to the sign of the world's curvature 

 we have, in natural units, (x K y K ) = cosh s or —cos s. 



Without insisting any further on these formulae let us only 

 draw from the last one this simple but interesting conse- 

 quence : — The shortest distance of two world-points being 

 manifestly an invariant, so is also x K y K . That is to say, in 

 passing from one to any other natural system (for in such 

 only we have the above Weierstrass coordinates with all 

 their simple properties), the sum of products of such co- 

 ordinates of two world-points retains its value. This simple 

 property, although arrived at by following upon the shortest 

 path from x to y 9 is at any rate independent thereof, and 

 belongs to that pair of world-points as such. This restricted 

 invariant f (% K y K ) which, by (27), can be written, with 



/ t 2 v 1/2 



^« = \ 1 + ]R») (^i + ^s + ^s + fs^-yy, • (32) 



must, of course, follow also from the group equations (26). 

 So, in fact, it does. As an instructive verification of the 

 above line of reasoning write the first five eqs. (26) for the 

 point x, and then for y, add the products of corresponding 

 pairs of coordinates and take account of the 15 conditions 

 between the a iK immediately derivable from those given in 

 (26). Then the result will be 



*;y.'=Vrf < 33 ) 



which was the property to be proved. This is the non- 

 homaloidal analogy of the invariance of the " scalar product " 

 of two four-vectors well-known from the older relativistic 

 vector algebra. The property (33) will follow even more 

 immediately by considering x K , y K as five- vectors in a five- 

 space, restricted by (17) to have the fixed " size " M, and by 

 remembering that the transformations in question are rota- 

 tions of the world-sphere or pseudosphere. With the aid of 

 (26) we can at once develop the whole vector algebra for a 

 non-homaloidal world, as an obvious generalization of the 

 older one. It is needless to show in detail how this is to be 

 done. We shall construct the entities of this kind together 

 with the rules of operating upon them every time these 



* The circumstance that the usual " s " is here replaced by is is due 

 to the negative sign in (18), which I retain for the sake of uniformity 

 with the present notation of most authors. 



t I. e. invariant with respect to the 10-parametric natural group of 

 transformations. 



