THEOKY OF POLYGON IC PROJECTIONS. 45 



We shall now prove analytically that any such oblique cone 

 that has one system of circular sections has also another 

 system of circular sections. If we have a cone passing 

 through the circle z = 0, x^ + y^ = a^, it will be a perfectly 

 general one if we take the apex at the point x =/, y = 0, 

 z = h in the plane y = 0, A line through this point is given 

 by the equations 



x—f=a(z — Ti) 

 y = ^(z^7i). 



This line intersects the plane z = in the point the coordi- 

 nates of which are 



Xi=f-ah 



y,= -^h. 



Since this point is to lie on the circle, we have 



(f-aJiY + ^^'^aK 

 But 



z — h 



z—h I 



By substituting these values we obtain 



(fz-hxy+TiY-fi'i^-f^y* 



This is the equation of a cone bearing the same relation to 

 the plane y = that the projecting cone bears to the plane 

 of the great circle. This equation may be written in the 

 form 



12 (^.2 + ^+ ^2 _ ^2) = ^[2fhx + (a^ -f + ¥)z - 2Jia'l 

 Hence, if the conical surface is cut by either of the planes, 



z = y 

 or 



2fhx + (a' -f + 7i')z - 2ha' = 5, 



the points of intersection will satisfy an equation of the 

 form 



x''-^y'' + z^^-2Ax^2Bz+D = 



