Manchester Memoirs, Vol. hi. (19 12), N'o. 4. 5 



Also Upq and Vqr are ultimately the osculating planes at 

 Q and R and the polar lines are perpendicular to them. 



.-../,,= UAV= UqV. 

 Now 



yds = Upa - Uqb = Upa + UqV- Vqb 



= df]- dx, as before. 



A comparison of either of the above demonstrations, 

 with the long analytical proof given by Laurent' will 

 show the great superiorit)- of the geometrical method. 

 In practice nearly all the results on geodesic torsion are 

 derived from Laurent's theorem, so that it is unnecessary 

 to give any further examples, although it should be 

 observed that most of them may be derived at once from 

 the definition by means of the spherical indicatrix. 



These views on geodesic torsion have been confirmed 

 by Mr, R. A. Herman, Fellow of Trinity College, Cam- 

 bridge, who points out that Laurent's theorem may also 

 be derived by the method of rotations. If the curve and 

 directions be referred to axes T, N and S, perpendicular 

 to N in the normal plane {Fig. i, 6" would be in NP 



produced so that NS = -^), then these axes move by a 



spin which is the resultant of spins represented by vectors 

 dx] along T', dO along B'a and —dx along T\ i.e., drj — dx 

 along T, ddsin x along N' and —dOcosx along S'. 



II. An Analytical Demonstration of 



Lancret's Theorem. 



An analytical proof of this theorem does not appear 



to have been given, probably owing to the fact that the 



geometrical proofs are so simple. The analytical proof, 



" Traits d Analyse^ tome vii., p. 25. 



