292 



LIPKA. 



(12) 

 we get 



(13) 



.i7 



dxi 



- a-;' 2 



i-r\r- = 1, 



dL 



dx. 



xL (1=1,2, ....,7l), 



as the differential equations of the natural family in euclidean space. 

 For any space, expanding (10), we get 



'^ddii 1 da 



ik \ / / 



i ik \oXk dxij h 



-—Z Oik Xi Xk —2:-— an Xi Xk 

 oxiik ikdXk 



Choosing the arc length as parameter along the curves, so that 



(4) 



S ttik x'i Xk = 1, 



ik 



writing L = log F, and introducing the Christoffel three-indices 

 symbol of the first kind ^ 



^ 1 / dan dak I _ «3«rt\ 

 \dxk dXi dxi / 



this becomes 



I' an x'i +2 KJ x'iXk 



% IK 



L^—anXiXk {1=1,2, ,71). 



ox I ikdXk 



To solve these equations for .r", we multiply by Ami, the minor of 

 Umi in the determinant a = | ox^^ | divided by a itself, and sum with 

 respect to /. From the properties of the determinant | flx^ | we have 



S an Ami = 0, if i ± m; 

 I 



(14) 



Our equations then become 



S an An = 1. 

 I 



(15) a-;; + S j ;,' j x'i xi, = S ^ (Ami - a-; .1- '), (m =1,2,...., n), 

 ik '^ ' I dxi 



where we have introduced the Christoffel symbol of the second kind 



ik ) r^^~l 



= S Ami \_l \ 



111 



Equations (15) reduce immediately to eciuations (13) if we place 



9 Bianchi, Geometria differenziale, vol. 1, chapt. II. 



