92 Professor Hamitton’s Third Supplement 
of containing these two known curves, and what other conditions are necessary, in 
order to oblige this surface to be the hyperboloid sought, let us employ the following 
general form for the equation of a surface of the second degree, 
Av’ + By + C2 + Day + Hyz+ Fzur+ Ga+ Hy +Iz+K=0, (Oke) 
and let us seek the relations which restrict the coefficients of this equation when the 
surface is obliged to contain the two known curves. The condition of containing the 
parabola (Z""), gives 
K=0, H= —ig, H=0,, C20, b= —+ i; (Vv") 
so that, by this condition alone, the general equation (U™) is reduced to the follow- 
ing form, 
z=qythty—— (G+ Fe+Dy+ Az). Ww) 
qYy +4ty —F , 
In order that this less general surface of the second degree, (JV), should contain 
the ellipse or hyperbola (S'"), it is necessary and sufficient that we should have the 
relations, 
G=-Ip, D=-Is,, A4=—-tf,: (X*) 
the general equation, therefore, of all those surfaces of the second degree which con- 
tain at once the two known curves (S”) (Z""), involves only one arbitrary coeffi- 
cient, and may be put under the form 
s=pxtqyttriv+sayt+hty +rrz. Os, 
This general equation, with the arbitrary coefficient X, belongs to the guiding parabo- 
loid removed, that is, to the surface (VV), when we suppose 
\=0; (2) 
and the same general equation belongs by (/?") to the sought hyperboloid (7"*), when 
ent ze (A) 
To put this last condition under a geometrical form, let us, as we have already consi- 
dered the intersections of the hyperboloid with the two rectangular co-ordinate planes 
of ay and yz, consider now its intersection with the third co-ordinate plane of xz, 
that is, with the plane of curvature (Q") of the given final ray. This intersection is 
the following hyperbola, . 
iNT 
y=0, c= +3r0'—4 5 22, (B") 
and the corresponding intersection for the surface ( Y"™) is 
y=0, 2=pr+ dre +022; (C4) 
the condition (4") is therefore equivalent to an expression of the coincidence of these 
two intersections; and if we oblige the surface of the second degree (U") to contain 
