ae 
Prof. Waterman on the Foci of Par 201 
sin(u-++v)=sini, and sin(w —v)=sin(¢+2v). Substituting these 
values, and we find for the general equation of intersection, 
y’?cos?v — 2az’sinv cosv sini +-a/*siné sin(é-+2v)=0. 
If we make i+-v=180° — », or ¢+-2v=180°, the plane AB will 
become parallel to the extreme element of the cone CF. Under 
this supposition sinz + sin2v, and sin(i+2v)=0. Introducing 
these values, and the equation of intersection becomes y’*cos*v — 
Qaz'sinv cosv sin2v=0. But sin2v=2sinvcosv: hence y’*cos*v 
—Aar'sin?vcos?v = 0; or y/? —Aaz’sin?v = 0; or finally, y’? = 
Aar'sin?v. But asinv=OD. Representing this by 7, we have 
y’? =4rsinv.z’, or, omitting the accents of the variables, y?= 
Arsinv.x, which is the equation of the common parabola. In 
this equation, 4rsinv is the parameter, and rsinv the distance 
from the vertex of the curve to the focus, which may be repre- 
sented by 4 Since v, and consequently the sine of v, is constant 
for the same cone, we may represent it by b. We shall then 
have y?=4brr. But since 6 is constant for the same cone, the 
distance from the vertex to the focus will vary with r. 
Making r=0, we find 5 =0. 
If mn (fig. 2,) be made equal to b= 
sinv, we shall have from similar trian- 
gles, Cnt tn 2: On’: tn’ 3: Cn’: tn”, 
&e.; and Cnt mni:Cn! 2 m’n’2 Cn’: 
m’'n!’, &c. Hence, int mniit/n! 3 m'n’:: 
tn”: mn", &e. But mm is the dis- A 
tance from the vertex to the focus of that parabola cut from the 
cone at the point where in=r=1. If the spaces Cn, nn’, n/n”, 
&c. be made equal, we shall have for én’ =2in, m'n’ =2mn, mn” 
=83mn, &c. But when tn=r is equal to zero, we have seen 
A A my 
ee 
that x=. Hence, 
P 
Ist. Whenr=0, 7=90, or the focus is at the vertex of the cone. 
Sxcoxp Senrizs, Vol. I, No. 2.—March, 1846. 26 
