52 Proceedings of the Royal Irish Academy. 



arbitrarily either sense of MN as the positive sense at the point of the curve 

 which we take as initial point. We then determine the positive sense of 

 MB by the condition that the trihedron (trirectangular system of axes) 

 M{TNB) is superposable on OXTZ by a movement of M[TNB) regarded as 

 a rigid body. 



As M moves along the curve, the trihedron M[TNB) moves in space. 

 The elementary movement is specified by a translation ds along MT, and 

 by rotations round MT and MB proportional respectively to the torsion and 

 the curvature. We take the torsion and curvature to have the same sign 

 as these rotations. The torsion, for instance, is positive if the trihedron 

 rotates round T in the sense {NB) as M advances along the curve. 



If the curve is discontinuous, or if we have to deal with separate curves, 

 we must make a new convention for each curve (or branch) as to the positive 

 sense of ds and the initial position of MN. 



36. We denote the du-ection cosines of the tangent, principal normal, 

 and binormal at a point [xyz) by ai/3i7i ; 02/3272 ; ai^rji- The torsion we 

 put equal to a or 1/r; the curvature to c or Ijp. 



With these conventions, Frenet's formulse are 



dai dui das 



ds ' ds 



with similar equations in (5 and j, 



ds ds ds 



Torsion of a Curve of a Liriear Complex. Prof. Mc Weenei/'s results. 



37. The two following theorems were communicated to the writer by 

 Prof. H. C. McWeeney, of University College, Dublin : — 



(a) TJie torsion is proportional to the sqvMre of the cosine of the angle 

 hetvjeen the binormal and the axis of the eomjilex. 



(b) If wxfy. denotes the perpendicular from the point \ of the curve on the 

 osculating plane at the pioint fx, then for any two points A and /a on a curve 

 of a linear complex, we shall have 



o-atV = (^mtVa.! (19) 



tvhere a/^ cviul a^ are the torsions ajt X and /j,. 



Since all the curves belonging to the same complex which pass through 

 a point, say A, have the same osculating plane and the same torsion at the 

 point, we may suppose A and fx to lie on two different curves of the same 

 complex, and theorem (b) will still hold. Theorem (b) is given, for the 

 particular case of a twisted cubic, by M. C. Servais [Mimoires couromiAs par 

 V Acadimie Boyale de Bruxelles, 1898). 



To prove these theorems, we take the axis of the complex as 2-axis. The 



