'50 KANSAS UNIVERSITY QUARTERLY. 



pencil from the node to the two points of intersection and the points 

 of contact is harmonic. Projecting this: — given three nodal cubics 

 having a common node and passing through five other fixed points; let 

 a conic be passed through the common node and two of the fixed 

 points, touching two of the cubics. The pencil from the common 

 node to the points of contact and the point where the conic cuts the 

 third cubic is harmonic. 



The following theorem may be proved in similar manner: — given a 

 system of cubics having a common node and passing through five other 

 fixed points; let a conic be drawn through the common node and two 

 of the fixed points; the lines drawn from the points where it cuts the 

 cubics to the common node form a pencil in involution. 



A variable chord drawn through a fixed point P to a conic subtends 

 a pencil in involution at any point O on the conic. Inverting from 

 O: — a system of circles through the double point of a nodal circular 

 cubic and any other fixed point P, is cut by the cubic in pairs of points 

 which determine at the node a pencil in involution. Projecting: — a 

 system of conies through the node of a unicursal cubic, two fixed 

 points on the curve, and any fourth fixed point, is cut by the cubic in 

 pairs of points which determine at the node a pencil in involution. 



We give another proof of the theorem that the three points of inflec- 

 tion of a nodal cubic lie on a right line. This is easily shown by 

 inversion and is a beautiful example of the method. 



There are three points on a conic whose osculating circles pass 

 through a given point on the conic; these three points lie on a circle 

 passing through the given point. * (Salmon's Conies, Art. 244, Ex. 5.) 

 By inverting from the given point and then projecting, we readily see 

 that there are three points of inflection on a nodal cubic which lie on 

 a right line. If the above conic be an ellipse, the three osculating 

 circles are all real; but if it be a hyperbola, one only is real. Hence 

 an acnodal cubic has three real points of inflection, while a crunodal 

 one has one real and two imaginary. 



The reciprocals of many of the theorems of this section are of 

 interest and will be given under Quartics. 



CUSPIDAL CUBICS. f 



Inverting the parabola from its vertex we obtain the Cissoid of 

 Diodes. The focus of the parabola inverts into a point on the cus- 

 pidal tangent which I shall call the focus of the cissoid. The circle 

 of curvature at the vertex of the parabola inverts into the asymptote 

 of the cissoid. This asymptote is also plainly the inflectional tangent, 



*See note A. 



tA few of the results of this section are due to the late Mr. H. B. Hall. 



