266 bepoet— 1879. 



point-co-ordinates by Professor Gay ley, 'On Cubic Cones and Curves' {Cambridge 

 Phil. Trans. 1856). If we write the parameter 



(& + 3)pqr + (p + q + r) (pi 2 + q 2 + r 2 - qr —pr —pq) = o. 



this assumes the canonical form 



p 3 + q 3 + r 3 + Qlpqr = o. 



The whole series of non-singular forms may be exhibited at once by line co- 

 ordinates. For order-cubics diagrams are most conveniently drawn by the 

 equivalent equation referred to the cusps (or inflexional points, dually viewed), and 

 the points in which the tangents at the cusps concur : 



(P + Q + R) 3 + 6kPQR = o. 

 where P = - 2lp + q + r : Q =p - 2lq + r : R =p + q — 2lr. 



4(1 -l) 3 

 K 1-21 + 4P 



The dual order-cubics are the two redundant hyperbolae, with three diameters, 

 simplex trilateral (Newton's Fig. 33) and simplex quadrilateral (Fig. 34). The 

 equiharmonic form, in which the stationary tangents or asymptotes concur', is drawn 

 (Fig. 42). The complex or bipartite form, in which an oval is enclosed by the 

 asymptotic triangle is not considered by Newton, but by his commentator, Stirling. 



In one case the form of conversion fails, when 



k=-|, (P + Q + R) 3 -27 PQR = o, orP* + Q* + R*=o. 



This represents part of the bounding curve when ABC is equilateral (supra, § 7). 

 But it is not represented in the canonical form, by the value 1= —h, except that 

 the line at infinity is common to both forms. Fspecial interest attaches itself to 

 this fault, since in piano there is thus occasioned a loss of one critical value, as 

 compared with spherics, which loss first occurring when the satellite is at the 

 centre of ABC, and therefore when it is within the bounding curve (P* + Q^ + R*), 

 continues throughout the various positions of the satellite. 



It may be noticed that the two harmonic curves of this group are conjugate, 

 i.e., each is the Hessian of the other. Hence it becomes apparent why the in- 

 variant (T = o) expresses the condition that the second Hessian shall be the original 

 curve. This relation holds good also when the parameters are imaginary. 



9. H three foci of a class-cubic be in any finite position, the envelope of the 

 stationary tangents of the inflexional cubics in a family of such confocal groups is 

 a class-quartic. 



If such a group be denoted — 



U= — = — f-i — +\p + uq + vr = o, 



apP + bq<4 + erIl r r * 



where P = ap — bq as C - «• cos B, and Q, R have similar values, so that 

 apP + bqQ, + crR = 2 (aY - ^cqr cos A) = 4A 2 . 

 One condition for a point of inflexion is 



d»U ffU _/ri»Uy» 

 dp* ' dq*~\dpdq) ~ ' 



This determines the envelope : 



(ap + bq + cr) ( — ap + bq + cr) (ap — bq + cr) (ap + bq — cr) 

 = 8abc pqr (ap cos A + bq cos B + cr cos C). 



Lines which join the centres of the inscribed and escribed circles with the foci and 

 the centre of the circumscribed circle, touch the envelope. 



This is the analogue of Pliicker's theorem : the locus of the cusps in a family of 

 groups of redundant hyperbolae is the maximum ellipse, which can be inscribed in 

 the triangle formed by the asymptotes. 



If the triangle formed by the foci is equilateral, the class-quartic degenerates 



