without the Equivalence Hypothesis, 117 



The iifferential equations of the geodesies or shortest lines 

 of the tour-dimensional world are immediately seen, by (18) 

 and (17), to assume in Weierstrass coordinates the simple 

 form 



^ 5 2 = jfi x *> K — ±,>>-o, . . . . (wOJ 



where 5 is measured along the geodesic itself. Needless to 

 say that these lines are of prime importance, first, because 

 — owing to their definition 8\ds = — they are invariant, and 

 then because they offer the first example of generally co- 

 variant laws, viz. the law of motion of a free particle. 

 Notice in passing that, in any coordinates u the eqs. of a 

 geodesic will be 



d 2 u t ( tc\ ) du K du\ _ n 

 ~d? + I ' J Is ~ds ~ ' 



the <7- , entering through the Christoffel symbols, being always 

 as in (4), since they are not moulded by gravitation or by 

 any other agent. i3ut let us return to the eqs. (28). Their 

 general integrals are 



x =a cos 



S) + *.»S> • • • < 29 > 



where cl k , b K are ten constants satisfying, by (17), the con- 

 ditions 



aa=-b K b=R\ and a^ = 0, . . . (30) 



to be summed according to the usual rule. These are the 

 equations of a geodesic of the world, for any constant 

 curvature. In natural units, 



x =a cosh 5 — 6 sinh s, for JR 2 >0, 



x k = a. cos s + ib K sin s, fori£ 2 <0. 



In order to find the shortest distance s between any two 

 world-points* whose Weierstrass coordinates are x K and z/ KJ 

 write (29) for the former and for the latter, multiply 

 them in pairs and add ; then, in virtue of (30), the result 

 will be 



is 1 

 G0 $ft= ^5 (*•#«)> ( 31 ) 



a well-known formula of multidimensional non-euclidean 

 * Not exceeding certain obvious limits. 



