54 



take (liis point as the orij^iii of a rectangular system of coordinates. 

 The equation of an arbitrary quadratic surface of revolution is: 

 f{x,y,z) = x' + y' -\-z'^(t {<tx -\-by^czy y2Ax-\-2Hy-\-2Cz-^ L>=0. 

 Tiie axis of revolution is defined by the e(|uations: 



0/ a/ " d /^ 



^X =: Öy ^=: ds 



abc 

 or 



x^ A y+B z -\- C 



"and passes thi'ough O if 



A_ B_ C_ 

 abc 

 Consequently only the axes of the surfaces 



^.» _[- y^ _^ 2^ _j_ ,f ^ax + 6// + czy + 2pf(a.i' -f èy -|- c^) -f y = 



pass through 0. 



We only consider 0* 's through the four given points {xi,yi,Zi), 

 hence : 



^^' 4- y.' 4- 2.' + « (Ö.IV + i^y. + cziY + 2/5 (a.t'. + by, + c-c .) + y = 0; 

 elimination of (/, j? and y gives : 



I -"^v' + ?a' + 2''' (^«'^-v + % + ^^0' ^^'^^ + %'■ + ^^/ 1 1 = 



As (7,, b, c are the direction cosines of an axis through 0, they 

 are proportional to the coordinates of an arbitrary point of such a 

 straight line. Consequently the equation of the complex cone of 

 becomes : 



I '^'i* + yi' + ^i' ('1'^*-^ + yyi + ^^^Y '''•^'' + ?///*• + ^^' 1 1 = o. 



In a similar way an equation may be derived defining the rays 

 of r in an arbitrary plane. 



^ 5. If the origin of a rectangular system is placed at the centre 

 of the sphere through Ai, the equation of the complex of rays in 

 line coordinates may be written: 



P4 'Ih Pe 



Py X, y, z, I 



A' .^\ y. z., 1 =0 



A' •^', y* z, 1 



^/ •'■. y. z. 1 



where Pi =r p^^v -|- p,y/+ p,^,-. 



