TRANSACTIONS OF SECTION A. 265 



5. These equations may be presented in a simpler form. Let P, Q, R, denote 



the several functions k 2 — 2k2/u — — , *2//3y — -^r^ftya, 2xa/3y — -7Tj5(2/3ya) 2 . 



r\r 2R 4K' i 



The invariants of § 4 may be written : 



P 2 + 12kQ = o. . P 3 + 18kPQ + 54 k 2 R = o. 



These may be combined to form two cubics in k : 



PQ + 9*R = o ... (1) 3PR = 4Q 2 . . . (2). 



If we neglect — and its powers, the direction of the asymptotes can be ob- 



XV 



tained by the resultant of two quadratics in k, and if the first power of — be also 



Si 



retained, the position of the asymptotes may also depend on the solution of two 

 quadratics. 



They are found to be eight in number, but only six real. Two more asymp- 

 totes would appear to be given by the factor 2/3y, which occurs in the eliminant. 

 But this factor is irrelevant, since Q = o, R = o satisfy the equations (1), (2), so that 

 4a/3yA — 2/3y sin A. 2//3y is a factor of the resultant, and should be omitted. 



6. Let the three foci of the cubic constitute the vertices of an equilateral 

 triangle. 



A group of confocal cubics is thus denoted : 



2<pqr + (xp + yq + z?-)(p 2 + q 2 + r 2 — qr —pr —pq) = o. 

 S = (* 2 - 2*A - 3A 2 ) 2 + 6k(< - 3A)2/3y = o, 



if each side of ABC be the unit of length. 



It is remarkable that k — 3A measures S, so that 



6Apqr + (xp + yq + =/') (p 2 + q 2 + r 2 — qr —pr —pq) = o, 



denotes an equiharmonic cubic, whatever be the position of the satellite. We can 

 examine its properties apart. Thus the Hessian of this family is the same complex, 

 wherever the satellite is placed, viz., the centre of ABO, and the line at infinity. 

 The species of equiharmonic is thus determined : for the Hessian of the other 

 species consists of three real points. 



Its Oayleyan is also independent of the satellite, and determines the line at 

 infinity and a point-conic at the centre. The evectant of T is also an equiharmonic 

 cubic of the other species, so that the series of equiharmonics may be multiplied 

 indefinitely. 



7. The bounding curve, when ABO are the vertices of an equilateral triangle 

 is a complex, one portion forming a tricuspidal bicircular quartic. 



(1) It is shown in § 6, that k — 3A measures S. This value substituted in T 

 gives the bicircular quartic 



(j3y + ya + a/3) 2 = 4aj3y(a + )3 + y) 

 Or, _ _ 



V/3y + s/ya + \/a/3 = o. 

 When transformed to line-co-ordinates it exhibits an acubitangential class-cubic 



2) i + q i + r* = o, 

 whose bitangent is the line at infinity. 



(2) The second factor of S is 



k 3 - k 2 A - k(5A 2 - 62/3y) - 3A 3 = o, 

 When combined with T 



4/c 2 2/3y + «(36a£y - 8A2/3y) + |(2/3y) 2 - 12A 2 2/3y = o. 



Their eliminant is the locus of the satellite, when the confocal family contains in- 

 flexional cubics. The direction of the asymptotes may be obtained by neglecting 

 A and its powers in these two equations, and their actual position, by retaining the 

 first power of A only. 



8. The group, in which the satellite is the centre of ABO, has been studied in 



