612 



n{n — 1) 

 always be written as a sum of — quadrates of complete diffe- 



n (n — 1) 

 rentials, ds^ can be reduced to a sum of — + 1 similar qua- 



drates. Hence the space A',; can be placed in this case in a eucli- 



dian space of 



n(n-l) (n-l)(n -2) 



2 ^ 2 



dimensions. As the number of degrees of freedom of the geodetic 



moving system amounts exactly to 



(n^l)(n-2) 



2 



the required proof has been furnished. 



If we now return to the case r=2, />j=:/), (j^=zq, the number 



p{p—\) q{q — 1) 

 of degrees of freedom is -|- — — . The quadratic form 



breaks u|) into two forms, which may be written as a sum of 



q{q^-\) p(p+l) 



— -^ resp. — quadrates. Therefore the space A„ can be 



placed within a euclidian space of 



p{p±}).q(qA-l) p{p-l) . q{q-l) . 

 — 2~" + ~2"~ = -^— ■^-2~" "^ " 



dimensions, and here again the required proof has been furnished. 



The case 9 > 2 may be reduced to the preceding one. For this 

 purpose the differential form is divided into two. The spaces of one 

 of the systems, say P,, then again contain themselves at Jeast two 

 perfectly perpendicular systems of pai-allel geodetic spaces. Then 

 the second part of the differential form is once more divided etc. 



If owing to the existence of the P;-systera the division of the 

 differential form is: 



2 2 2 2 2 3 



ds* = & ^, dx = ap ^. dx + &g ^. dx, 

 in which a^, and a, are the ideal radices of the two parts of the 

 fundamentaltensor a*, the differential equation of a geodetic line 

 will be : 



Sijj compossing itself only of the measure vectors e[ , . . . , e'p 



> > 



and a^ only of e^+i, . . . , e„, we have: 



