421 
Mr . Ivory on the astronomical refractions. 
of the curve placed at the eye of the observer ; and, at 
any determinate height, r is the angle contained by two tan- 
gents drawn from the extremities of the intercepted arc; or 
it is the sum of the angles which the two tangents make with 
the chord of the arc. When the curve is continued to the 
boundary of the atmosphere, or at least so high that the air 
has no longer power to deflect the light from its rectilineal 
course, the chord may be considered as parallel to the tan- 
gent at the remote extremity ; and then r is the astronomical 
refraction. The formula is perfectly general, and will apply 
in all hypotheses of density, since no particular relation is 
established between the variable quantities u and x. 
3 . But there are relations between the pressure of the air 
and its density and temperature, which must be attended to 
in the solution of this problem. Let p‘ and t denote the 
barometric pressure and the temperature on the centigrade 
scale at the surface of the earth, and put the same let- 
ters, without the accent, for the same things at the height x : 
then, if /3 = the expansion for one degree of the thermo- 
meter, we shall have 
* p ___ 1 4 - / 3 t f 
p—T+W x 7‘ 
In order to prove the truth of this formula, we may sup- 
pose a volume of air to be inclosed in a manometer, the 
pressure being p ' , the density p', and the temperature t': then, 
if the pressure be changed to p y the temperature remaining 
the same, the density will become, 
and, if the temperature be now likewise changed to r, the 
new density will be equal to 
