Frocet'dings of the lioyal Irish Academy 

 ••• AK) = 0, 



^.^^ = -^; 



AK) 



KB^ 



J {K) 

 Hence we have the equation of this quartic surface expressed in a 

 canonical form 



where 



^■/l) ■ '' = "■ 



We might consider particular cases of this quartic surface, accord- 

 ing as the conic becomes an ellipse, hyperbola, parabola, or circle. 



In particular, the imaginary circle at infinity. 



In this case, the plane w becomes the plane at infinity, and the 

 axes X, y, z rectangular; hence we may put, without introducing any 

 other peculiarity, f = = g = h, and a = /3 = y=l in the original 

 equation. 



And the Jacobian quadiics become of the fonn 



x^ + y- + 

 viz., spheres, where 

 =/(A) = 



2 (Lx + My + JVz) w 



«+ 2A 

 5 + 2X 

 





 



C + 2X 

 n 



+ Xw'^, 



I 



m 



n 



d-X^ 



The system of quadrics which have double contact with the surface 

 must also be spheres, since they pass through the circle ("imaginary) at 

 infinity, and the locus of the pole of the plane at infinity with 

 respect to them, that is, their centre, is 



w'' X y % 



X a+2 



y 3 + 2A 



z c-vIX 



= 0, 



