TRANSACTIONS OF THE SECTIONS. 17 



As this investigatioii has a double interest, it is desirable that the five cubic 

 cones of the third class should have distinctive names*. 



2. On the cubic referred to a triple cyclic arc. — Let the triple cyclic arc and its 

 satellite include any angle c ; since the third arc of reference is arbitrary, assume 

 it to be the quadrantal polar of the intersection of the other two. 



The trilinear equation to this group is 



Ka'+3i3(4;t=)=0, vrhere (4m=) denotes a=+/3^+cV+2ai3 cose, 



the expression for six times the volume of the plane tetrahedron formed by the 

 centre of the sphere and the vertices of the spherical triangle of reference. (The 

 symbols may denote the sines of the arcs in question.) It is seen that the cubics of 

 this group have a diametral arc and a Nevytonian centre at the point of inflexion. 



3. All cubics with trijile cyclic arcs have triple foci. — The equivalent tangential 

 equation to these cubics denotes in general curves of the sixth class, 



9»:V'(2=+r=)2+.32K(;;y+>-^ cos e)^— 36/c(j^' + >") (r+r-) (pq+r' cos c) 

 +12(p''+7-y(p-+q^+c-r^-—2pqcosc) = 0. 



This equation may be arranged to exhibit the triple focus 



(p—q cos c) {9K'(2pq cos c—p^—q-) (p—q cos c) —4iK(q—p cos c)'} 

 +(/>^+y^+cV— 2pg' cos c)wi=0. 



4. If the non-singular cubics of this group be complex, all six real foci are situated 

 on the dia7netral arc ; if simplex, only four. — After removing the factor which de- 

 notes the triple focus from the preceding tangential equation, the remaining factor 

 denotes three other foci, 



9k(2/}2 cose— ;;-— j')(p— g cose)— 4(5'— ^ cosc)^ 



Its discriminant will be found to be the same as that of the given cubic equation 

 to the curve. Hence follows the truth of the proposition. 



5. Critic centres in the general case. — Pliicker has defined them for plane cubic 

 curves as middle points, irrespective of their being the sites of nodes. 



A plane cubic of a particular group intersects the lines of reference in three col- 

 linear fixed points : the locus of the middle point of a straight line through one of 

 these points, intercepted between the other two lines of reference, is a hyperbola ; 

 the intersection of three such hyperbolae determines the critic centi-es. 



The same definition is applicable to spherical cubics. 



In particular, if {Ka^y=la-\-m^-\-7iy) denote a spherical cubic of the first group 

 refen-ed to three rectangular arcs of reference, the critic centres determined by this 

 definition are the intersections of three non-singular complex cubics with concur- 

 rent cyclic arcs, 



la JH/3 ily 



_ 6. Critic centres of cubics with triple cyclic arcs. — They are two in number, and 

 lie in the quadianta'l polar of the intersection of the point of inflexion. At a critic 

 centre, 



Hence the equation of § 2 jields these data, 



y=0 : Ka-+2afi+2^'' cosc=0 : a^+iafi cosc-i-3/3'=0. 

 It appears from the last equation, thus arranged, 



(a+2/3 co8c)-''-^=(l-4 sin^c)=0, 



* They might be distinguished as simplex or unipartite, complex or bipartite, veri- 

 bitangential, acubitangential, and inflexional. 



