Egan — Linear Complex, and a certain class of Ttvisted Curves. 5 1 



In this expansion Ar = (Or) + (rO), and hence, by section iv, equation 

 (13), (e), ^. = (r= 0,1, 2, 3, 4, 5). 



Hence Wi has a zero of order six for 9 = f. It follows that every 

 algebraic F-curve whose degree is less than six belongs to a linear complex. 



For consider Wi as a function of 9. Its order is the degree of the curve, 

 and therefore less than six. Since it has a zero of order six for 9 = t, it 

 must vanish identically. 



Independence of the Condition . (33) = 0. 



34. So far we have not proved that the identity (33) =0 is not 

 involved in the identity (22) = which characterizes a P-curve ; in other 

 words, we have not shown that a P-curve may exist whose tangents do not 

 belong to a linear complex. 



The locus of the point t" + &at\ t^ + 2bt\ P + 3bt\ t + a is such 

 a curve. In effect, the plane coordinates are easily found to be 



t + b, -5t^- 15at-, 5t' + IQaf, - t' ~ 6bt% 

 and (22) = 0, (33) = 360 (a - b). 



The curve will not belong to a linear complex unless a = b. [There are 

 cusps at and co, accounting for four of the six stationary points. There 

 are two inflexions given by (03) /t^ = t^ + 2t {a + b) + 6ab = 0, which will 

 coalesce if {a + bf = Gab. In the latter case the third stationary point 

 will be a cusp and not an undulation, since a P-sextic with an undulation 

 must belong to a linear complex (vil, par. 50).] 



It follows that the condition (22) = does not involve the condition 

 (33) = 0. 



VII. — Metrical Eesults. 



35. The signs to be attached to the curvature and the torsion at a point 

 on a twisted curve are variously determined by different writers. The 

 system we shall adopt is Darboux's. 



We take a fixed system of rectangular axes, 0{JCYZ). If a rigid body 

 has the point fixed and is rotated round OX, we consider the rotation 

 positive if the directed line, initially coincident with the positive sense of 

 Y, coincides with the positive sense of OZ after a rotation of amount 7r/2 , 

 we denote this positive sense of rotation by {YZ). The positive rotations 

 will therefore be (YZ), [ZX), [XY) round the three axes. 



Supposing the curve to be described by a moving point, the direction of 

 motion determines the positive direction of the tangent. If M be the point, 

 and MT, MN, MB be the tangent, principal normal, and binormal, we choose 



[7* 



