RoGKR^—Orikoff07ial Trajectories of a Congruence of Curves. Ill 



may be seen that their product is equal to the prochict of the principal 

 torsions. 



(Ji) A second gerieralization of tloc surface of centres might be investigated 

 similarly ; by considering the family of curves described by the centres of 

 principal curvature as the point P moves along its Il-family, the coordinates 

 of the two centres of curvature being given by 



where - is a root of 

 P 



and 



K = 0, 



'i = X + fit, )) = y t pm, I =- z + fju, 



P' P 

 ai h g I 

 h hi f VI 



K = 



g f ci n 

 I m n 

 As in ((/), there are two groups of n-faniilies described by 5, >), c,. 



'2f = hs + Ci, etc. 



r. — Further Generalizations of Dupin's Theorem, etc. 



The following is a second generalization of Dupin's theorem : — If three 

 n families are mutually ortJiogonal, the 7iecessary and sufficient coiidition that 

 their common curves should he curves of extreme curvature on the families to 

 which they helong is Ii = f = I3, i.e. the mean toi'sions are equal. This 

 follows from the formulae (1) and (2) (p. 96), combined with the principle 

 (p. 105) that the torsions for the directions of extreme curvature are equal 

 to i/. 



In particular, if one family lies on a one-parameter system of surfaces, the 

 same is true of the other's ; for if Ji vanishes, L and I3 also vanish. 



But a more general theorem is the following -.—The necessary and sufjieient 

 condition that a curve of inte7'seetion of IIi a7id lis shmdd he a curve of extreme 

 curvature on IIi, ivhe^^e the thi'ce systems are orthogonal, is I2 = I3. 



For since the curve is a direction of extreme curvature on Di, we have 



Ta Ti3 



and using 



''is T31 Tl:t Tii 721 



1 1 d9i, _ ^ 



712 721 dSls 



etc., 



L-^.L-^^I. 



-/..i, 



723 '■32 



731 



we have 



therefore 



/3 = 0. 



