40 C. H. DIX 
Distance from 5S to left vertex 
Vi 2H cosa 
= D,(T) = ——_—_—— (7-= ="), 
sin (a—v#) Vi 
Distance between center and vertex 
2 D,(T) —D.(T) =~ A(T) 
Distance from center to focus 
=eA(T)=C(T). 
In Figure 2, 
fC ay fed ee and Folie — ala But F.T)' — Fly’ =2C(T). 
So (F2/T1'—Fy'T;')/2=C(T). Also (Fo’I;’+Fi'Iy')/2=A(T). Hence 
e= (Fo/T)! — Fy Ty) / (FT +F yl’) = Sin d/sin a. 
We may thus write down the equation of the ellipses in the form 
cos a sind z 
| (r+ (ViT—H cos «| 
sin? a—sin? 3 
(ViT—2H ( I st I y 
cilia ud bois NGAI Huet =9) 
[v(T)}? 
ee Ta ae rae 
I I 4 sin? 3 
(V,T—2H cosa)? — (: ——— 
sin (a+) sin (a—#) sin? a 
which reduces to 
((sin? a—sin? 3) X(T)+cos a sind(ViT—H cos @))? 
sin? a 
+ (sin? a—sin? #)(Y(T))?=4 cos? 8(ViT — 2H cos a)?. 
Perhaps the best way to picture what is happening is to imagine 
the cone with a fixed axis moving up with a constant velocity so that 
the velocity of the vertex moving along this axis is 9,/cos a. Then the 
sections of this cone with the plane surface of the earth expand with 
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