TRANSACTIONS O?' THE SECTIONS. 17 



Oil Plane C'uhics of the Third CJuss iv'dh a Double and a B'i:i<jh Focm. 

 Bij IIe^-hy M. Jeffery, M.A. 



1. The classification of class-ciibics is simpler than of plane orcler-cubics, because 

 there are three real foci in each of the former, -whereas two of the asymptotes of 

 order-cubics may be imaginary. 



The three groups, arranged by the coincidence of the foci, have been stated m 

 the Transactions for 1875, and the third group of spherical curves there sketched. 

 This group (K}f = q) has been also fully considered in the 'Quarterly Journal _ of 

 Matheinatics' for 187G, both for plane and spherical class-cubics, -witli illustrative 

 dia'J'rams. These two memou-s contai»-a complete classification of circular cubics 

 by interpreting Boothian as Cartesian coordinates. 



"2. In the classification of the second group {Kp-q=r) the two foci will be con- 

 sidered fixed, -while the satellite-point varies. There is a certain quartic curve, the 

 locus of tlie satellite-point, when there is a point of inflexion in the curve; if the 

 satellite-point is within this curve, there -will be three critic linos ov bitangents ; if 

 it be on the curve, there will be one bitan<j-eut and two otlicrs coinciding in a sta- 

 tionary tangent ; if it fall beyond the bounding cuiTe, only one bitaugent is possible. 

 The critic lines or bitangents are the common tangents of three parabola, whoso 

 foci are severally the satellite-point and the two foci of the cubic. When these 

 parabolre are drawn, the bitangents are obtained graphically. 



The several cases of cubics o'f this group -will be next considered, according to the 

 position of the satellite with reference to the bounding curve -while all three points 

 are finite ; and subscc^uently, when the satellite-point and tlie.foci are, one or more, 

 at infinity ; also when the three points are collinear. 



3. The gi'oup may be thus represented : — 



K2yq-{-4-A'r = Q, 



where 'i£^-=a})V+bqQ,+cr'R; p, q, r are the current line coordinates, and F=(ip— 

 hq cos C-c;- cos B : Q, R have like values for points at infinity, as the quadrantal 

 poles of the sides of ABC. 



There are usually three critic lines, whose eq^uation.s are obtained by partial 

 differentiation : — 



apV = A^\l) : bqQ=2A-{2) : crPt=-2A-(3). 



Since the condition p = q = r satisfies all these equations, the critic lines must 

 touch three parabolse, whose foci are the vertices and whose axes are the perpen- 

 diculars drawn on the sides of the triangle of reference, and, httera recta, arc four 

 times, twice, and six times its corresponding altitudes. 



There are three of these critic lines, one of which is always real. 



4. The Cartesian equation to the bounding ciu've (§ 2), or locus of the satellite- 

 point, when there are stationary points in this group of class-cubics, is 



273/''-!-9i/'(2+4a;-f5.r')-fy^(l-l-a;)-(-l-10.r+9a;-)-.i--(l-f9a')(l+.r)^' = 0, 

 or 



{27z/'+I8/(x'-Hl)-(9.c-M)(.r+l)'}(y-+'") = 0- 



The double focus is the origin, and the focal distance on the x axis is taken as 

 unity ; a pair of asymptotes is inclined to the diameter at angles +30°. 



The envelop of the stationary tangent is a hyperbola, and of the single (one- 

 -vvith-twofold) bitangent is a cubic of division V., whose double focus is at in- 

 finity. 



5. Classification of the figures of class-cubics with double foci*. 



I. When the satellite-point lies beyond the bounding quartic, there is one bitan- 

 gent only. 



The triangle of reference formed by the foci and the satellite-point is taken to be 

 equilateral. 



* The diagrams illustrating the critical and the companion-ciu-Tes, according to these 

 nine capital divisions, -were exhibited at the Glasgow Meeting, and ore ready for piibli- 

 ciition. 



