290 . LIPKA. 



remaining constant; along such a curve the element of arc length 

 evidently takes the form 



(2) dsi = Van dxi. 



A direction passing out from a point Xi is determined by the ?i 

 direction cosines 



(3) ^^=^' (^= 1. 2,....,7i) 



which are related by the identity 



(4) i:aik^i^k=l. 



ik 



If ^i^^) and ^t^^^ determine two directions passing out from a point 

 Xi, the angle between them is given by 



(5) C05 CO = S Uik ?i(i) ^fc(2)^ 



ik 



and hence the angle between two parameter curves Xi and Xk is 

 (b) cos (jiik 



Van a 



kk 



If the space is euclidean, the parameter lines at any point may be 

 chosen as n mutually orthogonal straight lines, hence, from (6) and (2), 



(7) aik =0 if i ± k, and an = 1 (i, k = 1,2,. . . . , n) 

 and the element of arc length takes the form 



(8) ds'- = dxr" + dxo- + .... + dxn\ 



§5. Differential equations of a natural family of curves. 



Our problem is to find the differential equations of the curves for 

 which 



(9) t^ I F ds = I F VZ aik dxi dxk 



JiPo) Jlo ik 



minimum. 



The variation of this integral, keeping the end points Po and Pi fixed 

 must then be equal to zero. Using the usual symbol 5 as the symbol 

 of variation, we have 



