Rogers — Orthogonal Trajectories of a Congruence of Curves. 105 



The equation for determining the directions of zero torsion for general 

 axes is of course 



dx dy dz 



I m u = 



dl dm dn 

 combined with Idx + md// + ndz = 0. 



Eelation hetween Torsion ami Curvature. 



Let p/, p/ be the radii of curvature, and W, th' the radii of torsion of 

 normal curves touching the axes of x and y. Then, from the preceding, 



1 



1 



-K 



1 



— 7 = - «1. 



} ~ 



— > 



Pi 



p. 





Tl 



= - a. 



therefore, 



1 cos- e' /I l\ ■ a 

 - = 7- + — -, ,] sni 



P pi \ Tl Tx j 



1 cos'0' / 1 1 



cos I 



• sin W cos 0' + 



siii^' 



P-i 

 sin- 0' 



.Pi P'-) 

 9' being the angle which the normal curve makes with the axis of .r. 



Taking the directions of extreme curvatures for axes, and putting 6 for d', 

 («i, L, etc., changing with the axes). 



whei'e 



and for general axes 



1 cos' 6 sin- ti 

 = + , 



p fH P-. 



■""- = i/+f~--)sin0 COS0, 

 r " \fi P-J 



(3) 



/ = Z), - «, 



1 1 



- + - 



T y;./ (dn dm\ 

 [dy dz) 



Also for the special axes 



1-1 = 0- • / = A = 1. 



The equation (4) is an extension of Bonnet's well-known formula for 

 geodesic torsion, to which it reduces when /= 0. We verify what was proved 

 before that the sum of the normal torsions of two perpendicular directions is 

 constant at a point, being equal to \I, which we named the mean torsion at 

 the point. Tlia wean torsion is equal to the normal torsion for eitJicr direction of 



