Egan — Linear Complex, and a certain class of Twisted Curves. 57 



Metrical Properties of Algebraic F-C^irves. 



44. Lemma 1. — If A, aud n are any two points, one on each of two twisted 

 ciu'ves (or both on the same curve), then, with a proper convention of signs, 



where Si and -s, are the arcs of the cui-ves, measured from fixed points to X 

 and yu, i\u. is the mutual moment of the tangents at X and ji, a^ is the torsion 

 at jx, and t,^ is the perpendicular from X on the osculating plane at fi. 



Take the trihedron of tangent, principal normal, and biuormal at X as 

 axes, and let fi be {xyz). 



Then -n^ = aiJO + fizij + jzZ, -~ = a^ (ajS: + ^2}J + y-i^) 



0S2 



dx 

 (since Sos -r- = Sosai = 0). 



Hence ^logir^^ - a^ (a^: + j3zi/ + y2Z}/{a^c + ftiy + 732). 



jSTow transfer the axes to the trihedi'on at fx, and let X be (XYZ). Then 



a^c + jS.y + y.z = - Y, a^X + ^^y + y^z = - Z, — log tta^, = - rr^ Y j Z. 



082 



Differentiating with respect to s„ and remembering that o-^ is constant 

 for this differentiation, we get 



Y^-l-Z^— 

 log ttah = (7n — = rr^txij.lTT xij.- 



ds,ds2 ^ ^ "^ Z 



Gorolla,ry. — Since Sa^ = S^a, we have 



log — ^ = g^ -!i - __ 



45. Lemraa. 2. — On an algebraic curve, where the homogeneous co- 

 ordinates {x) and (a) are represented by functions of t, rational on the 

 <-plane or on a Eiemann siu'face, we have clearly 



nte = '^ai(e)x.{t)/Z(t)Jf{e), 



where Z(t) and (j)(f) are the results of substituting the functions Xi{t) and 

 ai{t) respectively in the left-hand members of the point equation of the 

 plane at infinity aud of the plane equation of the circle at infinity. In the 

 case of an algebraic P-curve where the fimctions (x) and (n) have been 

 chosen as in sections TV. and V., Z aud f are rational, of orders n and 2n 

 respectively, and have the same poles as (»,, a,). The order of these poles 

 is doubled in the case of ^. 



K.I. A. PBOC, VOL. XXIX., SECT. A. [8]. 



