48 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
upon the plane of the base the projection of the vertex shall fall at 
the orthocenter H. Now let the whole be orthogonally projected 
upon the plane of the base. 
~~ — 
a a 
From this conception it is seen that HA, HB, HC and HA’, 
HB', HC’ are projections of elements of the cone and HO the 
projection of the axis. Let the axis of the cone be bisected by 
planes parallel to the base. There are three such bisecting planes: 
one midway between apex and base, another below the base, and a 
third above the base. Finally let all these sections be orthogonally 
projected upon the plane of the base. 
From these conceptions it follows immediately that— 
(a.) These projections are circles whose diameters are, respect- 
ively, 2, #, 3, the diameter of the circumcircle. 
(b.) The centers of these circles F, F’, F”, all lie in the line HO, 
