200 Prof. Waterman on the Foci of Parabolas. 
Arr. V.—Foci of Parabolas ; by Georce Waterman, Jr., A. M., 
Professor of Mathematics in Miami University.* | 
The foci of all parabolas cut from a given cone by planes par- 
allel to the same element, are ina right line passing through the 
vertex of the cone. 
Let v, (fig. 1,) represent the angle. Fig. 1. 
OCD formed by any element of the — f 
cone with its axis: assume the axis 
of z as the axis of the cone; and 
let O be the origin of a system of | 
rectangular coérdinates. Represent- 
ing the distance OC by c, the gen- 
eral equation of the conic surface wil 
be of the form, z?-++-y?=(z—c)?. 
tang.?v. 
If the cone be intersected by a 
plane AB, perpendicular to the plane 4 ~ 
zz, and making an acute angle with - 
the axis z, it will cut out a curve, whose alviertins 26 will vary with 
the relative position of the plane and axis. 
Assume the point D, as the origin of a new system of coérdi- 
nates, and represent these coérdinates by 2”, y’ and 2’. Then for 
any point on the curve, as P’, we shall have PP=y=y. PD= 
x’, PS=-—z. Represent the angle which the cutting plane AB 
makes with the axis of z, by wu; the angle which the same plane 
makes with the extreme element CD by i; and the distance CD 
by a. We shall then have GD wiisihee: DS=r’sinu. PS= 
—z=v2'cosu. OS = «= OD-DS = asinv—-z'sinu ; and¢= 
acosv ; or t=asinv—z’sinu. y=y’, and z=—a2’cosu. Sub- 
stituting these values in the general equation of the conic sut- 
face above, and reducing, we obtain for the intersection, y/?cosv — 
2az'sinveososin(u + v) + 2*sin(u+v)sin(u—v) = 0. But since 
t+u+v=180°, ut+-v=180— —i,and u —v=180 — (¢+2v). Hence : 
* Since the following communication was received, Prof. W. has ra the 
appointment to the chair of Natural Philosophy and Astronomy in the Newton 
University, Baltimore; and ere this has probably entered upon its duties. 
