OF ARTS AND SCIENCES. 237 



live root of a double-sheeted one vanish, so as to leave the two actual 

 roots with opposite signs, the quadric degenerates into 



CjS^ftjP — CgS'rtgO = 1, 



an hyperbolic cylinder. In this case, the asymptotic cone is 



or 



Vc^SajP = ± VCjSwgP, 



a pair of real planes tangent to the cylinder at infinity. 

 When two roots vanish, 



CgSXe = 1 



represents two parallel planes, real or imaginary, according as the ac- 

 tual root is positive or negative. The asymptotic cone 



CgS^a.Q = 0, 

 or 



Sa^Q = 0, 



is a plane — which we might call a double plane — passing through 

 the origin and parallel to the pair of planes. These cases of degen- 

 eracy of quadrics we are about to study from an entirely different point 

 of view. 



Non-Central Quadrics. 



It has been already proved (page 224) that the centre of a quadric 

 is found by solving for d the equation 



cp,d + 7 = 0. 



Now the self-conjugate function may be treated under several differ- 

 ent forms, such as the rectangular, cyclic, ov focal (Ham. Elem., § 359, 

 I., III., and v.). Of these, I shall, for the sake of generality, consider 

 two, the rectangular and the cyclic, although the former is far more 

 convenient than the latter. 

 To solve the equation 



9o5 + 7 = 0, 



I shall use the general formula (Ham., §§ 347-350; Tait, Chap. V.) 



mQ = m(f~^y =. m'y — m"q)y -\- cp'^y. 



