610 



Now it follows from this mode of notation that they niay be 

 replaced by the invariant equation : 



r ^ s 2 V ^ (w, . T . . W9) = r ^ s '•: V ^ yw = 0, 

 or, as follows from the preceding, still more simplified without 

 making use of two auxiliary vectors: 



p 



2 y _ ^w = 0. ') 



But this equation is idenlically true, V^./W being a mulliple- 

 sum of isomers of 7,/W, and v,,w being zero. 



As the plane tangent-spaces of' pj dimensions, in the various 

 points of the spaces Fj have /^^-directions, which by means of geo- 

 detic motion pass into each other and in the invariant />^-direction 

 in 0, but never in any other direction, two spaces /^^ can Iheiefore 

 never intersect. A geodetic line in A'„, which has a linear element in 

 common with a detinite space Fj, is apparently altogether contained 

 in that space and in that space it is geodetic too. Hence two different 

 spaces Pj can never be tangent to one another. Thei-efore we call the 

 spaces Pj parallel ones. As any geodetic line having two points in 

 common with a l*j space, falls completely within that space, which 

 will be proved later on, we call a /j space geodetic. The ?• obtained 

 systems of parallel geodetic spaces P^, . . . , P, are at every point 

 of X„ perfectly perpendicular to each other. 



We shall first contemplate the case r = 2, ƒ;,=/>, j)^z=q. The 

 parameter-spaces ct\' n — 1 dimensions of the primitive variables x\ ... ,.r^, 

 are placed thus that each of I hem contains oo" /' ' spaces l\ those 

 of .<;/'+ 1, ..., .1" likewise with legard to the spaces Q. At every 

 point we place the mutually perfectly perpendicidar />- resp. (/-vectors 

 ^,v and fjw. The measure-vectors e\, x=: !,...,/> are then situated 

 in j,Y and for the measure-vectors q\„, ft = /> -|- 1, . . . , ?i the same 

 holds good with regard to ,,w. Because e'x ± e'„ we have 



x=:l, . . . , p 



ƒ! = ƒ> -I- 1 , . . . . , tJ 



hence the quadratic form ds^ may be written thus: 



ds* = ^ g,:, dx' dx' -\- 2 g,,y dx." dx" 



Now may be demonstrated, that <^y, is independent of .f/'+* ,..., .r", 

 and likewise ƒ/,,, is independent of ./•',..., .r>'. It is always possible 

 to choose a scalar k as function of x\ . . . , .i-", so that : 



^) This equation can also be obtained very easily by means of the direct analysis 

 used here. Another form of the same equation is : 



,w 1 (v 1 /-v) = 0. 



