I 



CoNJUN — Some Theorems on the Ttuisted Cubic. 255 



III. 



11. To examine more closely the connexion between the point and 

 the theory of the twisted cubic, it is necessary to establish the geometrical 

 relations of the points P and U (fig. 2) when the bitangent is at infinity. 



The rationalized equations of the quartic and conic are 



[yz + %x + xyf - 4:xyz (« + y + «) = 0, 

 and 



27 {yz + zx -^ xy) - ^{x + y + zy = 0, 



Let the equation of the conic referred to its principal axes be 



ft^ If ' 

 the equation of the quartic is found to be 



where a is the eccentric angle of one of the points at which the conic is 

 touched by the quartic. 



These equations are simplified by the substitution 



« = *"'"£ + i)' 



. fx . y 

 \a 



The equation of the quartic becomes 



{Kr,- 9y + 4 [^ + -n' - 27 + 9^rj} - 0, 

 and the conic ^») = 1. 



We can write ^ = 2^ - ,2 



2 

 and }] = — t' 



for a point on the quartic. 



The tangent to the quartic at t is 



, ^f 4 r,;; = 1 + t\ 



