RoGKRs — Orthogonal Trajectories of a Congruence of Curves. 9l) 



The torsion - of a curve (C) whose normal is /, m, n and whose tangent 



line at the point is perpendicular to the tangent line (a, /3, 7) of 6', is found by 

 interchanging- a, (3, j with A, /u, v, and changing the sign (since the orientation 

 of the tangent line, normal and bi-normal is altered*). Thus, 



- = a (O-iX + cu.jx + ciiv) + /3 (JiA + hifi + Jjv) + -y (c'„\ + c,^ + CiV.) 



T 



Hence, putting / = /ty - i'/3, m = va - \y, n = A|3 - fia, we have 

 1 1 7 A^w din\ I dl dn\ I dm dl 



T T \d]i dz I \dz dx) \ dx dy)' 



which is the formula in question. 



This gives a very definite geometrical meaning to the c^uantity /, whose 

 vanishing expresses the condition that the JT-family of ciu'ves may determine 

 a one-parameter family of surfaces. The sum of the normal torsions along two 

 perpendicular directions at a given point is constant and is equal to I. ^I may 

 therefore be described as the niean torsion of the I7-family at the point. 



More generally the magnitude of the component, along the direction of a vector^ 

 of the curl of the vector is equal to the magnitude of the rector, rnidtiplied by the 

 mean torsion of the Il-family defined hy 



Xdx + Yd;/ + Zdz = 

 where X, Y, Z are the magnitudes of the components of the vector at the 

 point. For 



where X^IR, Y= mR, Z = «i?. 



C. General Conformal Representation and Nok^iIal Torsion. 



The couformal representation of space on itself is expressed by a trans- 

 formation in which correspondhig directions at corresponding points ai-e 

 unaltered. Only inversion need be considered, since this and a repetition of 

 inversions are the only types in which the shapes of finite figures are altered, 

 the other tj'pes — similarity, rotation, translation, reflexion— being quite 

 simple. 



It is known that lines of curvature on a surface invert into lines of 

 curvature on the inver-se surface,f and this suggests the more general question. 

 What effect has inversion on the normal torsions of a il-family 1 



* See Salmon, op. cit., p. 3SS. 

 t 74iV., p. 409. 

 B.I. A. PROC, VOL. XXIX.. SECT. A. [14] 



