50 Proceedings of the Royal Irish Academy. 



32. Corollary 2. — If an algebraic curve is such tliat whenever a 'point A of 

 the curve lies on the osculating plane at a jyoint B, B also lies on the osculating 

 'plane at A, the curve 7nust belong to a linear cmnplex. 



This is the partial converse (since we only assert it of algebraic curves) 

 of a well-known property of curves of a linear complex. 



Consider an algebraic curve with this property. Its class is clearly equal 

 to its degree ; for the points at which the osculating plane at A meets the 

 curve are those whose osculating planes pass through A. 



Again, it is a P-curve. For consider a singularity A with the character- 

 istic numbers p, q, r, and let B be a point where the osculating plane at A 

 meets the curve again. If B is not on the tangent at A, the number of points 

 coincident with A in which the curve is met by the osculating plane at B is r 

 (the number of branches at A). Suppose a point |3 near B on the cui've. 

 The osculating plane at /3 meets the r branches of the curve in r points near 

 A, and the osculating plane at each of these passes through i3, by the 

 hypothesis. Let /3 move towards B : we see in the limiting case that the 

 number of osculating planes coincident with that at A which can be drawn 

 through B is r. But the number of these planes is precisely the class p) ' 1 

 of the singularity A. Hence p - q = r. 



If B lies on the tangent at A, a similar argument shows that 'p - r = q. 



That being so, consider the function 



S =^ad9)xi{t)l%ai{t)Xi{e). 



Considered as functions of 6, the numerator and the denominator have 

 the same zeros, namely Q = t (thrice) and the % - 3 other points which lie on 

 the osculating plane at t, and whose osculating planes pass therefore through t. 

 They have likewise the same poles ; namely the poles of the -Xt (which are also 

 those of the a,). 



Hence S is independent of 0. A similar argument shows that it is 

 independent of t. It is therefore a constant. To determine its value, let Q 

 tend towards t. Then 



2ai(e)a;..(0 — > (^ - 0' (30)/3 ! 

 2a.- (0 X, (6) — ^ (0 -ty (03)/3 ! 



Hence the value of S is (30)/(03) = - 1, and therefore Wi = 0. The rest 

 foUows by corollary 1. 



33. Corollary 3. — For an algebraic P-curve, putting B = t + (6 - t), and 

 expanding x^ (0) and a,- (6) in terms of d - t, we find that, if 6 is sufficiently 

 near t, 5 



w, = -2Ar(e-ty/ri + {e-tyc, 



where G is finite or zero for = t. 



