346 On atmospherical Refraciion. 
2; another mode of calculation must be pursued ; ‘the quantity 
2 necessarily becomes a divisor; and the whole expression ac- 
quires the factor ria which necessarily leads to the rule of 
t 
Mayer. 
Mayer’s formula was, no doubt, investigated by means of the 
hypothesis of a uniform decrease of density in ascending from the 
earth’s surface ; and, for the sake of further illustration, we may 
deduce it from the analysis in the Magazine for September, The 
hypothesis mentioned is contained in the equation fw =s, f 
being a constant quantity; for w is the decrease of density, and 
Ss, the height ascended divided by 2. If fw be substituted for 
s in the equation of the pressure, we shall obtain 
y we 
and, as this expression must vanish when w=1, we get f= 2. 
At the boundary of the atmosphere, the equation s = 2 takes 
place, so that the total height is equal to 27 .* 
It must be observed, however, that in a limited atmosphere, 
the ultimate density, 1 —w, is not, strictly speaking, evanescent; 
but equal to some sinall quantity, less than what would obtain 
at an equal height in an atmosphere of uniform temperature. 
Now, if 2w be substituted for s in the formula for the refrac- 
tion, we get 
Pe Bsin A dw 
~~ gfcost A+ (4i—m2 Bw 
To integrate¢his expression, assume 
w=(l—e)u+eru: 
then, A being the radical quantity in the denominator, we have 
A= wo feos? A+ (4i—28) (1 —e*) w + (4i—26) e u%} 
and, by determining e? so as to make the expression on the right- 
hand side a square, 
A4i—FB te, 
cos A TT Jerse? ? 
A= cosA + eu /4i—28”° 
dw = 2edu 
A oy v 4imsp 
: 2p 
dr =sinA x x edu, 
v 4i—28 
Now, w and z increase together from zero to 1; wherefore 
oO 
is ad x sin A xe: 
A 4t-- 26 
© Mecanique Céleste, vol. iv. p. 260. 
