Mancliestcr Memoirs, ]\)I. Ivi. (191 2\ No. 4. 3 



extremity. Hence if 7 represents the geodesic torsion 

 we have, b\' M. I^onnet's definition, 



yds = N'N, = BX' - BA\ = BN' - BN, 



which proves the equivalence of the two definitions. 



As a verification it maj' be noted that the new 

 definition gives the correct result for a line of curvature 

 or a geodesic. This is obviously so in the former case 

 s,mcG BN=BN':- In the latter case 



yds^P'B-PB^BB\ 



since N coincides with P and N' with P' . Hence the 

 geodesic torsion is equal to the ordinary torsion, a known 

 result which justifies the use of the term. The definition 

 includes as a particular case the definition of a line of 

 curvature given in the former paper referred to. 



The most important result in this subject is Laurent's 



M Y 



theorem, the analytical statement of which is y = P — ~, 



^ being the complement of the angle between the oscu- 

 lating plane of the curve and the tangent plane to the 

 surface at any point and /-* the ordinary torsion of the 

 curve. This is easily proved by means of the spherical 

 indicatrix. We have 



yds = BN' - BN= BB' + B'N' - BN 



^dj]-d^, 



y = P-^^. 

 ds 



The same result may also be obtained by elementary 

 geometry. Let PQ, QR, RS be three consecutive and 

 equal elements of the curves, p, q and r their middle 



- loc ciL, p. 68. 



