TRANSACTIONS OF SECTION A. 491 



Their common tangents are seven by a well-known theorem (Salmon's ' Higher 

 Algebra,' Art. 248.) 



4. A moveable point (P) lies on AD, arcs (or in piano, right lines) connect 

 P with two other points B, C : the envelope of the bisector of the angle BPC is one 

 of the preceding cubics. Similarly, the bisectors of the angles AQC, BRA, if Q, R 

 move on BD, CD, envelope the other cubics, which determine the critic lines by 

 their mutual combination. This is an extension of a theorem of Pliicker. 



5. In the most general case, where ABC may have any position on the sphere, 

 the critic lines are thus determined : — 



op = ffg = V 



- apP + bqQ + crR ~ apP - bgQ, + e?-R apP + 6jQ - crR' 

 where P= ap - bq cos C - cr cos B, and Q, R have similar values. But the equation 

 and form of the locus of the satellite, when there are inflexional values in the 

 group of class-cubics, is not here determined. 



6. By reciprocating, the critic centres of a group of spherical order-cubics are 

 determined to be seven by the intersecting cubics — 



pa 



6V - 2aa (aa + 6/3 cos c + cycos, 1) 



In piano, as is well known, these degenerate into three critic centres, formed 

 by the intersection of three hyperbolae. For then cos a=cos 6 = cos c = l : 

 6V = 2(aV + 26c/3y cos a) = 4 A*, and the cubics become hyperbolae : 



pa g/3 ry 



— Oa + bd-h cy~ rta — 6/3 + cy ~ aa + 6/3 — Cy 



7. On a Cubic Surface referred to a Pentad of Co-tangential Points. 

 By Henry M. Jeffery, M.A. 



1. A cubic surface may be generated as the locus of the foci in involution of 

 all the transversals, concurrent in a fixed point, which meet a system of quadrics or 

 conicoids, intersecting in a quadro-quadric curve. This is an extension of Cremona's 

 method of generating plane cubics to solid geometry. Dr. Salmon's process leads 

 to the same analytical expression. In such a system of conicoids, the locus of the 

 conic curves of contact of enveloping cones, with a common vertex, is a cubic 

 surface. 



2. All the pole planes of the fixed point, with respect to the conicoids, intersect 

 in a straight line PQ, which is one of the 27 lines on the cubic. If the system of 

 conicoids be referred to their self-conjugate tetrahedron, and if the fixed point be 

 E, the centre of the inscribed sphere (1,1,1,1), the five triple tangent-planes through 

 PQ touch the cubic in the four vertices of the tetrahedron and in E the centre. 

 Consider any pair of lines, as AP, AQ, forming a triangle APQ with PQ. Then, 

 beside the original plane APQ through each of the lines AP, AQ, four more triple 

 tangent planes can be drawn : in all nine such planes. The same is true of the 

 planes through the pairs of lines at B,C,D,E, which constitute with PQ triple 

 tangent planes. Thus the 45 planes are exhibited. Again, besides the line PQ and 

 the five pairs of intersecting lines, which meet PQ in ten points, there are 16 lines, 

 which may be determined in five different ways. Eour of the five triple tangents 

 through each of the lines AP, AQ (exclusive of the common plane APQ), deter- 

 mine 8 lines each, the number required. The same process may be used with the 

 same results, if the triple planes through the lines intersecting in B,C,D,E be used. 

 The arrangement of these planes and lines, whose discovery by Professors Cayley 

 and Salmon constituted an epoch in Solid Geometry, may be compared for simplicity 

 with Professor Schafli's double-sixers, and Dr. Hart's cubical system of grouping. 



3. The analogues to Maclaurin's theorem on tetrads do not present themselves. 

 But points on the cubic may be thus multiplied. 



Transversals through a fixed point P on the cubic pass through the vertices 

 A,B,C,D,E, which constitute the pentad of co-tangential points, and intersect the 



