Consequently 



«11 



Vke, . . . Q/j — (Vk) (Ci . . . C/,) -\- k 2 ^ {&. Cx) (e, . . . e._i a e.+i . . . e^,) = 0. 



■/ 



By complete transvection with: 



C/, . . . . Cx+i e« Cx— 1 . • . • Ci Cv 

 all the terms except the {yi -\- l)-{h give zero, hence: 



d \ ü = 1, . . . . , p 



üy 



= 



Now 

 and 



thus 





a^ a^ji ==: a^v «v. 





^vx ö!u.j 



= ttfj^y, ttv + a, J, Ovx = W. 



Hence the linear element in the Q spaces is independent of 

 x'^, . . . , .vP ; the corresponding property of the P spaces i-elative 

 to ,tP~^^, .... .r" is similarly demonstrated. 



This property can^^lso be expressed in the following manner. 



II. J f in a general space Xn is placed a system of co'^^i' parallel 

 geodetical spaces of p dimensions P., having perfectly perpendicular 

 to it a system of qpp similar spaces Q of n — p dimensions, a figure 

 in a definite P-space loill be congruently projected by the Q-s paces 

 on all the other P-spaces. 



For p=l this is the well-known property that the distance of two 

 definite Q-spaces measured along the P-lines is constant. So we can 

 here introduce in this case for primitive variable xMhe curve length 

 measured along these lines from a definite space, the spaces remain- 

 ing parameter-spaces. Hence the linear element may then be expressed 

 in the following way : 



« 2,...,u 



ds^ =. dx^ -4- 2 . g^-^ dxi^ dx" 



in which the <7^v do not depend on x^ . '). 



As, however, a quadratic differential form in n — 1 variables can 



^) This formula has already been derived by T. Levi Givita. Nozione di parallelismo 

 in una varieta qualunque e consequente specificazione geometrica della curvatiira 

 Riemanniana. Rend, di Pal. 4i2 (17). 



40 



Proceedings Royal Acad. Amsterdam. Vol. XXI. 



