430 
Mr. Ivory on the astronomical refractions. 
refraction ; the French mathematicians make it equal to ~ ; 
and the result found above falls between these limits. 
6 . In the expression of the refraction (C), the quantities 
i and a are very small fractions ; and cos.® 9 varies from i to o 
as the zenith distance increases from o to 90°. For a con- 
siderable extent from the zenith cos. e 9 will greatly exceed 
i and « ; and so long as this is the case, we may find the 
value of r by expanding the radical quantity in a series. 
Proceeding in this manner, and retaining only the two first 
terms of the expansion, we shall get 
dr=cc( 1 +*) Tan. — j : 
and, by integrating from u = 0 to &> = 1 , 
r=* Tan. fix { i + g — * } , 
the terms multiplied by a, i u, <x being alone retained. Now 
s ( l — u) — J d 5(1 — a>) — J s du\ 
and because 5(1 — u ) = o, both when on = 0 and u = 1 , if we 
take the whole integrals between these limits, we get 
f s du=f ds ( 1 — a ). 
But f ds ( 1 — w) between the limits u = 0 and u = 1 has the 
same value that J — ds ( 1 — u) has, between the limits u = 1 
and u = 0 ; and this last integral is equal to the whole pres- 
sure, or to unit : wherefore 
f s da =2= 1 ; 
and, by substitution, 
r=*Tan. 0 x|i + 
By means of this formula, which was first found by Laplace, 
the French tables of refraction are computed as far as 74° 
from the zenith. The quantities i and a, depending only upon 
