Mr. Ivory on the astronomical refractions. 435 
To find the refractions near the zenith, we must develope 
the radical quantities and retain, as formerly, such terms only 
as are multiplied by a, az, a 9 : in this manner we get, 
r= x Tan.0 • { 1 + cjjrj + f Tan.* 5 
° r > . , 
r =“ Tan.flx {i + “ — 
and this is no other than Laplace’s formula, which is thus 
deducible from the most simple, as well as the most compli- 
cated, hypothesis. 
The same formula may be thus written, viz. 
r == « ( 1 "T a | Tan. 0 + a y xa( 1 «) Tan. 0 x Cos>a 0 1 ; 
and, the second term of this expression being inconsiderable 
in comparison of the first, we get, for an approximate value, 
r=«( i + a) Tan. 9 ; and again, if we substitute r for 
a ( 1 + u ) Tan. 9 we shall obtain, 
i — 4 a 
U ~ *(! + «)*’ 
r = a(i + a) - |Tan. 9 — nr 
But this value of r is no other than the two first terms of the 
developement of a ( i -f- a ) Tan. (0 — nr) ; and hence, 
r = «( i + a) Tan. (0 — nr), 
an expression which must be considered as an approximation 
of the same order with the formula of Laplace, and it must 
be restricted within the same limits. It is to be observed, 
however, that the two forms of expression will not be en- 
tirely equivalent unless the same values of a and i be, in every 
case, substituted in both ; which implies that n will vary a 
little according to the pressure and temperature of the air. 
The formula for the refractions near the zenith is common 
