OF ARTS AND SCIENCES. 249 



Such a sphere will be represented by the equation 



0^ = 2nS«i(?, 

 for this has 



S«i() = 



for a tangent plane at the origin. Its intersection with the parabo- 

 loid is given by 



a cone referred to its vertex. The cone will represent a pair of planes, 



if 



— k 



^2 — — ; 



for then 



c^{S\q -\- (.2) 4- CjS^agO = 0. 



Now if we reduce to Cartesian co-ordinates by the substitution 



Q = xa^-{- ya^ -\- 



3' 



we find that 



c^iy" — x^ — y" — z"-) + C322 =0, 



— c,(x'' -]- z") = — c^z"" ', 

 and finally 



Since we can always take c^ <^Cii — whatever these roots may be, — 

 the equation just found will be imaginary if c, and c^ have opposite signs. 

 The geometrical interpretation of wliich is that an hyperbolic parabo- 

 loid can have no circular section : a fact almost self-evident. 

 Let us substitute in the equation 



Ccpc = (Cg c^z , 



the cyclic values of c^, c^, and Cg (Ham. Elem., § 357, XXI.), 



c^ — ^1 = TAjU -\- SAjW 



In the present case, c^ = ; and therefore 

 „ Tam — Sv o 1 — cos <' 



z- 



TAmH-Sam" i + cos<; ~ '"^/* 



