Mr. Ivory on the astronomical refractions. 419 
trajectory from the points where the light enters into, and 
passes out of, the spherical shell of air ; then, if y represent 
the latter of these two lines, y + dy will be equal to the 
former. The distance of the intersection of the two tangents 
from the centre of the earth being a -f- x, V (a +•£ ) 2 — f 1 will 
be the distance of the perpendiculars from the same in- 
tersection ; and, on a circle described with the radius 
V(a-t-x) 2 — y\, dy is the arc that subtends the small angle 
contained by the two tangents ; wherefore, if dr denote the 
measure of the small angle, we shall have 
dy = drx.V{a + Jcy—y *; 
and, 
dr = - 1 =M=- 
V {a + xy— y x 
Again ; because the light is continually deflected in a direc- 
tion tending to the centre of the earth, equal areas will be 
described round that centre in equal times by the motion in 
the trajectory ; but the areas described in equal times are 
proportional to the velocities multiplied by the perpendiculars 
falling upon the tangents from the centre of forces : where- 
fore, the product v xy will have always the same magnitude 
at every point of the curve. Let v' be the velocity of the 
/ 
light at the surface of the earth, or at the point where the 
trajectory enters the eye of the observer ; and put y' for the 
perpendicular upon the tangent drawn from the same point 
of the curve : then, 
v xjy = t/xy 
Suppose also that 0 is the apparent zenith distance of the star, 
or the angle which the last-mentioned tangent makes with 
