. 
82 
inclined at an ¿0 or 180°- 0 to the plane of xy, and hence 
the surface may be generated by a variable circle in two 
different ways; but it will be observed that in every case, 
if the surface become one of revolution, the two positions 
coincide in one which is parallel to the two equal axes. 
105. Next, let the surface not have a center, and let 
its equation be 
By 2 + Cx 2 = 2A'oc. 
Then as before, if h, l, denote the co-ordinates of the center 
of the circular section, its equation reckoned from that point 
as origin will be 
By 2 + C (a?'sin 0 + l) 2 = 2 A'(jv cos 0 + h), 
with the conditions 
B = C sin 2 0, C sin 0 , l = A' cos 6; 
/* a 
sin 0 — i \/ — , and l = ^ cot 0. 
Hence the locus of the centers of the circular sections is a 
straight line parallel to the axis of the surface, i. e. it is a 
diameter of the surface; also B and C have the same sign and 
B < G; hence the surface is the elliptic paraboloid, and the 
cutting plane is perpendicular to that principal section of the 
surface whose latus rectum is the least. 
In the case of the hyperbolic paraboloid, since B and C 
have different signs, no plane can be drawn so as to intersect 
it in a circle. 
106. Any two circular sections of a surface of the second 
order, provided they be not parallel to one another, are 
situated on the same sphere. 
Since all circular sections are perpendicular to the same 
principal plane, let (fig. 53) represent that principal plane,