RoGKRS — Orthogonal Trajectories of a Congruence of Curves, 97 



But this evidently follows from the equations 



rs3 



i = o,i-i = o,i 



1 1 



— + — 



1 1 



— + — 



0,1. i 



'■31 '■32 



= 0, 



0, — + — = 0, — + — = 0. 



Tja ri3 Tai T23 



The normal torsions reduce to geodesic torsions, and therefore the 

 corresponding curves are directions of lines of curvature on each surface 

 through the point. 



Darboux' reciprocal theorem also follows very simply. If Hi, Ui represent 

 two mutually orthogonal systems of surfaces, then Ii= Ii = 0, and 6,2 is a right 

 angle. Let Da represent the n -family of curves which are the orthogonal 

 trajectories of the congruence of curves in which surfaces of Hi cut surfaces 

 of W* Then 6^^, ^3, are right angles ; hence the first five of the preceding 

 six equations hold, and the sixth is replaced by 



1 1 T 



T3I Tz-i 



The theorem now amounts to the statement that if 



1 = 0, 



T12 

 i.e. if the curves of intersection of surfaces of the systems 111 and ITa are 

 lines of curvature on the surfaces of 11., then /s = 0; that is to say 

 113 represents a family of surfaces, and Fli, fla. Da a triply orthogonal system. 

 The proof is algebraically obvious. 



Proof of Equation (1). 



If a, j3, 7 are the direction-cosines of the tangent-line to a curve, 

 I, m, n those of the normal and \, fi, v those of the bi-normal, then, by 

 the Frenet-Serret formulaef or otherwise, 



= X 



T 



cU 



df 

 ds 



dm 



ds ds 



dn 

 ds 



m 



dm 

 ds 



7 



n 



dn 

 ds 



* ris is defined by [mvii) dx + («i&) dy + (Zitna) dz = 0. 



t Salmon's " Geometry of Three Dimensions," 5th ed., vol. i, p. 387. 



