Rogers — Orthogonal Trajectories of a Congruence of Curves. 115 



The Unit-vector. 



When always l' + m'^ + n'^ = 1, we have a system of unit- vectors; and it 

 is evident that the Divergence (cf. p. 106) and the Curl are geometrically 

 definable by means of the n -family 



Idx + mdy + ndz = 0. 



Let u, V, w be magnitude of the components of the Curl of the unit- 

 vector (I, in, n). Then 



dii dm 



11 = -^ r- , etc., 



dy dz 



using the special coordinates (p. 101), u = -h^, v = a^, w = hy- a^ = /. 

 Thus the magnitude of the normal component of the Curl is equal to twice 

 the mean torsion, as we have already seen. To determine the geometrical 

 meanings of a^ and 63, we must investigate the curvature of the orthogonal 

 trajectory of the Il-family, this trajectory belonging to the congruence 



dx _ dy dz 

 I m 11 



Let I', m', n be the direction-cosines of the principal normal of the 

 orthogonal trajectory, s' and p' its^arc and radius of curvature. 

 Since the direction-cosines of the tangent line are I, in, n, 



dl 

 n' = 0. • 





I' 



f 



P 



dl dl dl 

 = —, = 1-- + m -- 

 ds dx dy 



and similarly 





111' dm ^ 



p' = ds' = ^- ' 



Hence, 





m' V 



,(, = -_, v = - 



P P 



ia = I, 



and therefore I'li + m'v + n'xu = 0, i.e., the direction of the Cuii of the 

 unit-vector is' parallel to the pi'incipal tangent plane of the orthogonal tra- 

 jectory of the Yi- family, it makes with the normal to the IV- family an angle 

 j^ where 



tan Y = 



i 



7 



p 



1 1' 



- + - 



ri T2 



and its magnitude is l__ + [ — ^- — 



'P 



R.I. A. PEOC, VOL. XXIX., SECT. A. [16] 



