Rogers — Orthogonal Trajectories of a Congruence of Curves. 101 



respectively, and subtracting the sum from the first cohimn, the second, 

 third, and fourth members of the first cohimn vanislr, and 



r=A, = - 



dx dji dz 



I III n 



dl dm dn 



The last written determinant is equal to — . Thus 



T 



_ 1 _ l_fds^ _ J_ _ ^ 

 Ti r-T\dsJ r^-T T 



If k, instead of unity, is the radius of inversion, -p- = — = — and there- 



dSi Ti ic~ 



fore the normal torsion, on the inverse family, of the in\'erse curve at the 

 inverse point is equal to the normal torsion, on the original family, of the 

 original curve at the original point, divided by minus the linear magnifi- 

 cation. (It should be remembered that normal curves do not invert into 

 normal curves.) 



Taking into account the simpler cases of conformal representation, it is 

 easy to see that for any conformal representation of U-familics, the new 

 normal torsion is equal to the old divided hy the linear magnificaMon, the plus 

 or minus sign being taken according as the representation is ' direct ' or 

 'inverse.'* 



A corollary is that curves of zero normal torsion invert into curves of 

 zero normal torsion on the inverse family, and in particular lines of 

 curvature on a surface invert into lines of curvature on the inverse surface. 

 It will be seen that there are two curves of zero normal torsion through each 

 point in space, and these may be either real or imaginary. 



D. ToESioN AKu Curvature uf Norjial Curves. 



The investigation of the differential properties of the normal curves 

 passing through a given point P in space, may be simplified (as for surfaces) 

 by taking P for origin and the axis of z for the n -normal. Then, near the 

 origin, neglecting higher powers of x, y, z, 



I = a^x + a-^y + a^, m = 5i* + 6,7/ + iiz, n = 1 , 



the coefificients of *■, y, z in n disappearing since the differentials of P + m^ + n\ 



* ' Inverse ' representation occnrs in an odd number of inversions or reflexions witli regard to ii 

 point or plane, ' direct ' in translation, rotation, and shrinkage or expansion. 



[14*] 



