162 Philippine Journal of Science 1920 
curve through the atmosphere and is tangent to the earth's sur- 
face immediately under 7; but its initial and final directions 
intersect at 7, at height p above the earth; the ray passes under 
Z at a height H above the earth; h is then the geometrical alti- 
tude of the sun’s upper limb (here negative, a depression). 
O 
Fig. 2. Geometrical relations. 
Clockwise arrows indicate positive angles, a, 8, y; the arrow 
for h, a negative angle, is counterclockwise. — 
From the figure 
28—=180°—2r, y=2r+90°h, a+B-+y=180°; 
whence 
B=90° —f, y=90°+h-+2r, a (h+r) : 
and the angles of the triangle are determined. Of the sides 
none is given, but 
(R-+-H) /(R+p) =sin B/sin y, 
(1+H/R)/(1+-p/r)=cos r/cos (h-+-2r). 
Now it is evident that the straight line TV is nowhere in 
contact With the earth; the perpendicular to it from the center 
ls greater than the radius, or 
or 
