38 CH DIX 
constructing the two greatest spheres tangent to the cone and to the 
plane (surface of the earth) on both sides of this plane. The two points 
of contact of the spheres with the plane are the foci. The case where 
v is greater than or equal to a need not be considered although the 
treatment is much the same as for the case ?<a. 
In Fig. 2 we show the construction of the foci. The cross-section 
there shown is perpendicular to the strike of the refracting interface. 
S; and S; are the two spheres. Evidently their centers are on the axis 
of the cone. The center C; of S; is also on the line bisecting the angle 
SI,'P. The center C2 of Sz is on the line bisecting the angle Li’J,’S. 
Hence F; is the foot of the perpendicular dropped from C; to JI’ 
and F, is the foot of the perpendicular dropped from C; to IJ’ As 
the conical wave front proceeds (with the normal velocity Vi) these 
spheres expand and the foci move outwards but the ellipses al] have 
the same eccentricity. The semi-minor axis of the ellipse is b. Then 
b=V (I,'I2'/2)?—(FiF2/2)2. The eccentricity is e=FiF2/I,'I2’.The 
points J,’ and J,’ move outwards with the velocities V;/sin (2+) and 
Vi/sin (a— x—), respectively. The distance from S to the center of the 
ellipse is Io’S— Ty'Te!/2 which is ms= T,'S/2—I2’S/2=(iS= Ty'S)/2 
So the velocity of the center is 

I V1 v1 ) 
; 2 = (a—vd) sin (a+?) 
which is a constant. The foci also move with constant velocities. The 
distances of the foci, the center, and the vertices J,’ and J,’ from S 
can readily be computed using the constancy of the velocities. In fact 
the distance of the center from S is 
DAT) =0(T—k) 
where & is such that when T=k, D.(T) =o. k can readily be computed. 
When D.(T) =o the vertex of the cone coincides with S. Fig. 3 shows 
the diagram with which k is computed. One can see directly that 
H 
k=——_ (1++ cos 2a) 
Vi cose 
(1+cos? a—sin? a) 
V,cosa@ 
2H 
Siren CON Ye oF 
1 
282 
