Mr. Ivory on the astronomical refraction. 
481 
thesis of Kramp ; but, as it is too complicated for calculation, 
he deduces from it this more simple value, viz. 
1 ■ — U — C 
by retaining only the part of the expansion of the function 
in the index that contains the first power of s. 
In all this Kramp is followed by Bessel, whose aim is to 
determine the value of e that will best represent all the ob- 
servations of Dr. Bradley, without paying any regard to the 
terrestrial phenomena, or to any farther theoretical consi- 
derations whatever. 
Now, there is an essential distinction between the rigorous 
expression of the density, and the approximate value used 
instead of it. The latter belongs to a finite atmosphere, and 
the former to one of unlimited extent. To prove this, we 
need only substitute c~ ^ ^ 5 for 1 — u in the equation, 
P =f—ds (1 — ») ; 
and then we shall get 
P = . 
1 £ 
the constant quantity being necessary, because P = 1 when 
s = 0. But as P cannot be negative, we have P == 0 at the 
top of the atmosphere ; and the total height will therefore be 
determined by the equation 
c -(i— )*-_ e==0 . 
If we could suppose that s is a very small fraction, and the 
height of the atmosphere very great, what has just been 
observed would be of little consequence. But, at the surface 
of the earth, we ought to have ^ = 1 — s = ±, and b = ± ; 
which would limit the atmosphere to about double the height 
MDCCCXXIII. 
s£Q 
