48 KAN'^AS UNIVERSITY QUARTERLY. 



Since the above transformation is rational, it follows that there is a 

 (i, i) correspondence between the conic and the cubic. Tliis fact is 

 also evident from the nature of the method of inversion. The cubic 

 has its maximum number of double points, viz: one; and hence is 

 unicursal. This unicursal circular cubic may be projected into the 

 most general form of unicursal cubic; the cuspidal variety, however, 

 always remaining cuspidal. 



By applying the method of inversion to many of the well known 

 theorems of conies, new theorems are obtained for unicursal, circular 

 cubics. If one of these new theorems states a projective property, it 

 may at once by the method of projection be extended to all unicursal 

 cubics. Examples will be given below. 



The following method of generating a unicursal cubic is often useful. 

 Given two projective pencils of rays having their vertices at A and B; 

 the locus of the intersection of corresponding rays is a conic through 

 A and B. Invert the whole system from A. The pencil through A 

 remains as a whole unchanged, while the pencil through B inverts into 

 a system of co-axial circles through A and B, and the generated conic 

 becomes a circular cubic through A and B, having a node at A. Now 

 project the whole figure and we have the following: — given a system of 

 conies through four fixed points and a pencil of rays projective with it 

 and having its vertex at one of the fixed points, the locus of the inter- 

 section of corresponding elements of the two systems is a unicursal 

 cubic, having its node at the vertex of the pencil, and passing through 

 the three other fixed points. 



Unicursal cubics are divisible into two distinct varieties, nodal and 

 cuspidal. The nodal variety is a curve of the fourth class and has 

 three points of inflection, one of which is always real. The cuspidal 

 variety is of the third class and has one point of inflection (Salmon, 

 H. P. C, Art. 147). Each of these varieties forms a group projective 

 within itself; that is to say, any nodal cubic may be projected into 

 every other possible nodal cubic, and the same is true with regard to 

 the cuspidal. But a nodal cubic can not be projected into a cuspidal 

 and vice versa. 



In applying this method of investigation to the various forms of 

 unicursal cubics and quartics, only a limited number of theorems are 

 given in each case. It will be at once evident that many more theorems 

 might be added, but enough are given in each case to illustrate the 

 method and show the range of its application. It is not necessary to 

 work out all the details, as this paper is intended to be suggestive 

 rather than exhaustive. 



