TRANSACTIONS OP THE SECTIOXS. ]9 



VIII. If the double focus and satellite-poiut arc both at iufinity. 



There is au iuflt-xional forui iu all cases, as appears from the equation lo the 

 svstem : — 



The reciprocal is a cusped cubic. 



IX. If the double and single focus are both <at infinity. 



Tlie lino at infinity is an acii-bi tangent in all cases, as is shown by the ec[uation 

 to the system : — 



G. To find the asymptotes of this group of class-cubics. 



Since the polar-point of au asymptote (p, q, r) lies on it, and also is at an infinite 

 distance, the coordinates of an asymptote must satisfy two equations : — 



(^ = X/(?+4AV=0, (1) 



and that to the polar conic of the line at infinity 



one of whose foci, as we should anticipate, is the double focus. 



The four asymptotes touch a conic, whose foci are p=0, 2q-\-p = 0. Hence also 



r{2q+p)=pq (3) 



The elimination of r from (2) and (.3) is a quartic equation. Hence there cannot 

 be more than four asyniptotes. Its discriminant is a factor of the discriminant of 

 the ternary cubic (1). Two imaginary asymptotes always connect the double focus 

 ■with tlie circular points at infinity. 



If the satellite (which is always on the curve) be at infinity, its connector with 

 the double focus is au asjnnptote. The extremities of two asymptotes may coincide 

 in a bitaugent. 



7. The centre or polar point of the line at infinity is on AB, the connector of the 

 foci, at the distance ^ AB from A, the double focus. AC touches the cubic in C ; 

 BC touches it where it meets the line {^—Ky=0). 



On Spherical Class-cubies iv'ith Double Foci and Double Cyclic Arcs. 

 By Henet M. Jeffery, M.A. 



1. This group may be denoted by line coordinates, as in platw : — 



^p'q+(G\yr=0, 

 where 



(GY y-=2(aY—2bcqr cos A), 



and the coordinates p, q, r denote the sines of the perpendiculars from the vertices 

 ABC on a tangent arc ; and generally the symbols may denote the sines of arcs. 



There are four critic values of the parameter and fom- bitaugential values. By 

 partial differentiation, for a critic value, 



2Kpq+2raV = : Kp'+2rbq=0 : (Q\)-+2rcU=0, 



where F = ap—bq cos G-cr cos B ; and similar expressions denote Q, R ; the line- 

 coordinates of a tangent are referred to the polar triangle as one of reference. 

 These conditions for a bitangent may be thus written :— 



apV=(6\y : 26jQ= (6V)= : 2c/-R= -(6V)^ 



These equations denote three spherical ellipses, whose foci are in the several cases 

 the points of reference A, B, C, and the corresponding points of reference of the 

 polar triangle of ABC. 



