52 KANSAS UNIVERSITY QUARTERLY. 



the cubic, and intersecting in a fourth point such that the two tangents 

 to the conies at their point of intersection, together with the two lines 

 from it to B and C, form a harmonic pencil; the locus of all such 

 intersections is a conic through B, C, and the cusp having the point of 

 inflection and the cuspidal tangent for pole and polar. 



(5) Reciprocating (4) we have: — through any point on the cuspidal 

 tangent draw the two other tangents, B and C, to the cubic. Touch- 

 ing B; C, and the inflectional tangent draw two conies, such that the 

 points of contact of their common tangent, together with the points 

 where their common tangent cuts the tangents B and C, form a har- 

 monic range; the envelope of such common tangents is a conic having 

 the cuspidal tangent and the point of inflection for polar and pole. 



(6) Projecting (2) we obtain the following: — through the point of 

 inflection draw any line cutting the curve in B and C; take any other 

 two points on the cubic such that the pencil from the cusp, O, O (B 

 P C Q) is harmonic; the conic passing through O B P C Q will pass 

 through the intersection of the cuspidal and inflectional tangents. 



(7) Reciprocating (6): — from any point on the cuspidal tangent 

 draw two other tangents, B and C, to the cubic; take any two other 

 tangents, P and Q, such that the range cut from the inflectional tangent 

 by B, C, P, Q, is harmonic; the conic touching B, C, P, Q, and the 

 inflectional tangent will also touch the line joining the point of inflec- 

 tion and the cusp. 



(8) Projecting (3): — through the point of inflection draw any line 

 cutting the cubic in B and C; through the cusp O and the points B 

 and C on the cubic and any other fixed point P, three^ and only three, 

 conies can be passed, such that the tangent to the conic and cubic at 

 their remaining point of intersection, together with the lines from it to 

 B and C, form a harmonic pencil; these three points of intersection 

 are collinear. 



SYSTEMS OF CUBICS THROUGH NINE POINTS. 



Let U and V be the equations of, two given cubics, then U + kV is 

 the equation of a system of cubics through their nine points of inter- 

 section. Twelve cubics of this system are unicursal, and the twelve 

 nodes are called the twelve critic centres of the system. (See Salmon's 

 H. P. C, Art. 190.) 



Let the equation of the system be written briefly 



a + ka^ + (b+kbi) x + (c+kCj) y + u^+Ug^o; 

 one, and only one, value of k makes the absolute term vanish; hence 

 one, and only one, curve of the system passes through the origin, 

 which may be any point in the plane. Make the equation of the 

 system homogeneous by means of z, and differentiate twice with 



