OP ARTS AND SCIENCES. 235 



Transfer the origin to a, and this becomes 



(^Q(pQ -f~ 2 Sog)« -(- S«(jp« — 1) (S«gia — 1) — (Sp(jp« -|- S«g)a — 1)^ 

 = S(>qp(>(S«g)a — 1) — (St»g)«)^ = 0. 



This equation represents a cone referred to its centre, because every 

 term contains q to tlie second degree. It is, then, the tangent cone from «• 

 If in the equation of the polar plane 



S(j(p« = 1 



a vanish, no finite value of q will satisfy the equation, and the polar 

 plane of the origin is seen to lie at infinity. The tangent cone from 

 the origin must therefore meet the quadric at infinity, and becomes 

 what is called the asymptotic cone. The equation of this cone is read- 

 ily obtained by substituting 



« = 

 in that of the tangent (ione 



S^qp(j(S«g)« — 1) — (Sp(jpa)'^ = 0, 

 which gives 



The rectangular transformation for this is 



and it can only be satisfied by real finite values of q, where one or 

 more c's are negative, and the quadric an hyperholoid. In the case of 

 the real or imaginary ellipsoid, the c's are all of the same sign, and the 

 asymptotic cone is reduced to its vertex, for its equation is only satis- 

 fied by 



t; = 0. 



The reality of any cone depends of course, in the same way, on the 

 difference of the signs of the roots of gio. Any cone may, then, be 

 regarded as the limiting case of an hyperboloid which is degenerating 

 into its own asymptotic cone. 



This idea leads us to consider how one form of surface of the second" 

 order may, by the modification of the constants in its equation, pass 

 by imperceptible degrees into some different form. 



When any root of a central quadric vanishes, the surface becomes 

 indeterminate in the direction of that root, and thus degenerates into a 

 cylinder. If c^, for instance, vanishes, we have 



