480 REPORT — 1 880. 



There is a single asymptote \ + a^ = o, wlien ^ = <x. No points of inflexion, 

 distinct from the cusps, satisfy the condition 



(Pk dK _ drX (h _ 

 dj' d$ d$' ^ ~ "■ 



7. By the aid of this bounding quintic allquartics maybe exhibited, which hava 

 a quadruple focus and a satellite-circle. 



If in such a group of §5, 8X = d^, 27a^K. = 2, so that (k, X) is a ceratoid cusp 

 on the bounding quintic, the quartic 



|.«Y + (I «^p - 2«| + l) '?^ + f (9«^ + 2) (l - U = o 



is inflexional. 



In a family of such groups, the locus of the point of inflexion is the hyperbola 

 (108kX = 1). 



For values of (k, X) on points in the neighbourhood of the cusp, the quartics are 

 veriban tangential with two cusps of the cardioid type, or acubitangential, as 

 (k, X) is situated on one or other of the branches which meet at the cusp. For points 

 within this space, the quartics consist of an oval, pierced by the hyperbolic branches 

 of another non-stapete or smooth oval. For points beyond this space the quartics 

 are bistapete with four, two, or no asymptotes, and also become smooth according 

 to the position of (k, X). It may suffice here to state, that for other critical values 

 of (k, X), one or other negative, the quartics are lima^onoid, i.e. unistapete in the 

 nascent form, or have bicusped bi-tangents, the reciprocals of biflecnoids, i.e., are 

 bistapete in the nascent state. 



8. Non-singular quartics may change their stapetes, without passing through 

 critical values; the stapete-points of transition are determined by aid of the 

 Hessian of the group, or by means of the invariants S, T, equated to zero. 



■ Let the centre of the satellite circle be at infinity. Such a group 



has no critical bitangential quartics, but its stapetes vary with X. The Hessian of 

 the group is 



(4X- + 6X + 2) ^* + (8X^ + 8X - I) f ,,^ + (iX^ + 2X) r,* = o. 



The real values of these points, which constitute the Hessian, and of the coincident 

 stapete-points, depend upon the auxiliary quadratic 



32X2 + 32X = 1, 



whose roots are -03033 and - 1-03033. 



Other transition-values of X are 0, — -6, —1. 

 If X = '03 or — 1-03 the quartics have fom- stapete-points. 

 X = — -5, there are two stapete-points. 

 \ = or — 1 , there is a tacnode at infinity, 

 or the quadric is bistapete in its nascent state. 



For intermediate or external values the quartics are quadristapete, bistapete, 

 or nonstapete. 



9. This slight sketch may suffice to explain the plan of this chapter of Plane 

 and a corresponding chapter iu Spherical Class-Quartics, which, it is hoped, may 

 shortly find a place in the ' Quarterly Journal of Mathematics,' illustrated by the 

 necessary diagrams. 



11. On the equations to the real and to the imaginary directrices and latera 

 recta of the general conic (a,h,c,e,f,g,h) {x,yiy = o ; with a note on a 

 property of the director circle. By Professor R. W. Genese, M.A. 



Let M ^ a or' + 2hxy + by^ + 2gn + 2/y + c = o be the equation to a conic referred 

 to rectangular axes : let (a,^) be the coordinates of a focus, x coad + y sin 6 = jy 

 the corresponding directrix, and e the excentricity of the conic. The equation to 

 the conic may therefore be written 



(x - ay + (y-^y = e' (x coa6 + y sin 6 - pY 



