MancJu'sfer Miiiioir:;, ]'ol. hi. (191 2), Xo. 4. 



IV. A Geometrical Treatment of Geodesic Torsion, 



By Lancklot v. Mkadowcroi- r, B.A., M.Sc. 



{Comniiinicated by R. /'.' Gwyther, AA/1.) 

 {Received October J I si, r()ir. Read NoTcmhii- i^tli, i<jii.\ 



In the first section of this paper is given an entirely 

 geometrical treatment of the subject of geodesic torsion, 

 the method adopted being one previously employed in 

 demonstrating several theorems on lines on surfaces.' 

 The remainder of the paper is devoted to sonic miscel- 

 laneous notes on lines on surfaces. 



I. A Geometrical Treatment of Geodesic 

 Torsion. 



The conception of geodesic torsion is due to M. Bonnet, 

 who defined it, under the name of second geodesic curva- 

 ture, as the ratio borne to the element of the arc by the 

 angle which the normal to the surface at one extremity 

 makes with the plane containing the element and the 

 normal at the other extremit)-. This definition does not 

 lend itself easily to analytical expression or to geomet- 

 rical treatment, and for the latter purpose at any rate 

 is not so convenient as one involving only angles between 

 lines, e.g., the normals to the surface and the binormals of 

 the curve. For these reasons I replace it by the following 

 definition : — The ratio borne to the element of the arc by 

 the difference between the angles made by the normals 



' Quarterly Journal of Rure and Applied Mathematics {i()Q()), No. 161. 



A/arch 6th, igi2. 



