2 Meadowcroft, Treatment of Geodesic Torsioit. 



to the surface at its extremities with the corresponding 

 binormal (or osculating plane). The equivalence of the 

 two definitions may be demonstrated by elementary 

 geometry in the ordinary wa}', but the following proof by 

 means of the spherical indicatrix is more satisfactory and 

 convincmg. Draw a unit sphere with O as centre, and 

 from O draw lines parallel to the tangent, principal 

 normal and binormal at one extremity of the arc ds, 

 meeting the surface of the sphere in 1\ P and B respec- 

 tively {Fig. i). Let T\ P', B' be their positions for the 



J'k- 1. 



other extremity. The resulting figure is known as the 

 spherical indicatrix, and it is a well-known property that 

 rr is tangential to TP' at T and BB' to B'P' at B'. 

 Draw ON, ON' parallel to the two normals to the surface. 

 iV obviously lies in BP since TN=hir. Similarly N' lies 

 in B'P'. Draw the great circle through T' and N and 

 let it meet B'P' in .V Let TT' = dO and BB'=^dr\. 



Now ON' is parallel to the normal at one extremity 

 of the arc and the plane OT'N is parallel to the plane 

 containing the element of arc and the normal at the other 



