28 report— 1877. 



a 1 (.r 2 +y 2 ) 3 (* 2 -27y 2 )-66Vy-(.r' 2 4-y 2 )=(2^4-% 2 )+36'y l ( 1 ^'-5G.ry-% 1 ) 



(In this equation ,r 2 is written to denote x 2 — b'-.) 



The curve is trinodal at the foci, and unipartite, with a pair of asymptotes 

 through the origin, and the (y) axis for a third asymptote. If the satellite lie on 

 either of the exterior loops of the bounding curve, there are four critic values, one 

 of them inflexional ; if the satellite fall within, five ; if without, three critic values : 

 if it lie on an interior loop, two critic values, one of them inflexional ; if within, only 

 one critic value, viz. the veribitangential. 



For positions of the satellite along the (a?) axis, the preceding quintic becomes 



(J 2 £-.r)(& 2 £ 2 -2*!+l) 2 =0. 



The first value resolves the cubic into a point and a parabola ; the other pair of 

 coincident values yields two bitangential and not inflexional cubics — four, if the 

 satellite lie beyond the foci. 



5. The case in which the focus at infinity has neither of the preceding positions, 

 and all the cases where the foci are finite, remain for consideration. 



On a Cubic Curve referred to a Tetrad of corresponding Points. 

 By Henry M. Jeffery, M.A. 



1. A non-singular cubic consists of an infinite number of such tetrads ; and if 

 any one be drawn, the remaining tetrads may be obtained by transversals drawn 

 from the several points of the cubic. Each tetrad ABCD yields three pairs of 

 dyads, AB, CD ; AC, BD ; AD, BC ; and each pair of dyad iines touches a sepa- 

 rate Cayleyan class-cubic. 



Let transversals through P on the cubic pass through A, B, C, D, and intersect 

 it again in A', B', C', D' ; then, as has been stated, these points constitute a new 

 tetrad; and if A', B', C, D' be joined in pairs with A, B, C, D, their points of 

 intersection are three, and with P constitute a new tetrad. 



These characteristic theorems, which are not new, maybe simply obtained by the 

 following proofs. It will also appear that the locus of the intersections of pairs of 

 tangents drawn at the points where transversals through a fixed point on the 

 cubic intersect it is a trinodal quartic, which passes through the first tetrad of 

 corresponding points, the three other corresponding points of the second tetrad of 

 which the fixed point is one, and through the coresidual or its own tangential point. 

 The loci of the several points in which these concurrent transversals are divided — 

 (1) Arithmetically, (2) Geometrically, (3) Harmonically — are (1) a crunodal cruci- 

 form curve (Newton's Species 7 or 8), whose asymptotes, three or one, are parallel 

 to those of the primitive cubic, and at whose node the rectangular tangents are 

 parallel to the asymptotes of its pole conic ; (2) a central cubic, with asymptotes 

 also parallel to those of the primitive cubic (Newton's Species 27) ; and (3) the 

 pole conic, as is well known, a rectangular hyperbola. 



2. If a system of conies be described 2. If a system of conies be inscribed 

 about four common points, and if in four straight lines, and if from each 

 through a fixed point a series of trans- point of a fixed line pairs of tangents be 

 versals be drawn, the locus of the foci drawn to the conies, the envelop of the 

 of the points in involution is a cubic. focal lines of the system in involution 



is a class-cubic. 



This theorem (which is due to Prof. Cremona) is equally applicable to Plane and 

 Spherical Geometry. Let the triangle of reference, ABC, be formed by the 

 diagonal points of the quadrilateral constituted by the vertices ; their coordinates 

 are ±x, ±y, + z ; and if the transversals concur in (/; g, h), the locus of the foci in 

 involution will be denoted by the cubic, 



£(# ~ Yff) + 'i(yf-ah) + S -(ag-ff) =0. 

 a /3 7 



