newson: unicursal curves by method of inversion. 69 



When nine of the basal points of a system of quartics coincide, the 

 quartics have a common triple point. This is nicely shown by invert- 

 ing a system of nodal cubics from the common node. The inverse 

 curves form a system of quartics having a triple point and passing 

 through seven other fixed points. The common triple point on two 

 quartics counts for nine points of intersection and the seven others 

 make the requisite sixteen. From our knowledge of a system of cubics 

 having a common node it is readily inferred that three of the quartics 

 must each break up into a nodal cubic and a right line through the 

 node. If the seven fixed points of the system of quartics lie on a cubic 

 having a node at the common triple point, the system of quartics then 

 consists of this cubic and a pencil of lines through the node. If two 

 of the seven fixed points lie on a line through the common triple point, 

 the system of quartics then consists of this right line and a system of 

 cubics through the other five points and having a common node at the 

 common triple point. 



The system of cubics having a common node may have one, two, or 

 three of the other basal points at infinity; and these may be all dis- 

 tinct or two or three of them coincident. Whence we infer that if the 

 system of quartics have ten coincident basal points, one of the tan- 

 gents at the triple point is common to all the quartics of the system. 

 If eleven basal points coincide, two of the triple-point tangents are 

 common to all the quartics. If twelve coincide, all three triple-point 

 tangents are common. These triple-point tangents may be all distinct, 

 two coincident, or all three coincident. 



If thirteen basal points coincide, the system of quartics then consists 

 of the three fixed lines joining the multiple point to the other three, to- 

 gether with a pencil of lines through the multiple point. If fourteen 

 points coincide, two lines are fixed and these with any two lines of the 

 pencil form a quartic of the system. If fifteen points coincide, only 

 one line is fixed and each quartic consists of this line and any other 

 three of the pencil. When all sixteen points coincide, any four lines 

 through it form a quartic of the system. 



In this paper cubic and quartic curves only are considered. I expect 

 in a future paper to extend the methods herein developed to curves of 

 still higher degrees. Many of the present results can be generalized 

 and stated for a unicursal curve of the nth degree. I have purposely 

 omitted all consideration of focal properties of these curves. There 

 are also many special forms of interest which do not properly belong 

 to a general treatment of the subject. 



