Smyly — A Note on Certain Curves, etc. 



373 



In a system of quartic cyclides wMch. have six common concurrent 

 double normals with the same points of bisection, the locus of the 

 middle points of the twenty-four other double normals is a twisted 

 cubic, which may also be regarded as the locus of the vertices of the 

 common self- conjugate tetrahedra of a fixed quadric and a system of 

 concentric spheres. 



The coordinates of any point on the twisted cubic may be written 

 in the form 



of hg ch 



~ a + A' ~ 2i"TX' ~ c + A ' 



to this point corresponds the focal quadric 

 o;^ y^ z^ 



+ 



+ 



a + k b + X c + \ 



1 = 0: 



from the point and quadric an infinite system of cyclides connected 

 with a twisted cubic is derived; the equations of this curve are 

 given by 



af hg 



X + 



ch 



a + X 



b +\ 



c + \ 



a + \ b + X c + X 



which immediately reduces to 



^+f y + 9 2 + ^ 



- 0, 



= 



hence, there is a doubly infinite system of cyclides connected with the 

 same twisted cubic. 



If P and Q be any two points on the cubic, the polar plane of P 

 with regard to the quadric corresponding to Q is identical with the 

 polar plane of Q with regard to the quadric corresponding to P ; 

 hence, given any point P and a quadric P, the quadric corresponding 

 to any point on the cubic obtained from them can be constructed 

 geometrically. 



Also, if any point be taken on the cubic, a quadric can be deter- 

 mined such that the locus of the vertices of the common self-conjugate 

 tetrahedra of the quadric and a system of spheres having the point as 

 centre, is the cubic ; the system of quadrics so obtained is confocal. 



