344 On aimospherical Refraction. 
If r denote the refraction of the star, the angle of incidence 
of the ray of light will be + 7; wherefore, supposing that | + 
to 1 is the constant ratio of the sine of incidence to the sine of 
refraction, we shall have 
Sin (9+7) = (146) sin 4, 
Sin ¢ cos r+ sinr cos ¢ = (1+) sin ¢, 
Sin r = 6 tan $+ (1— cos7r) tan ¢. 
It will be abundantly accurate, even at the horizon, to put 
rs ~ 2 
dir a for sin r, and 4 7? for 1— cos 7; and then by seeking 
the yalue of 7 from the resulting expression we shall get 
g tan>¢ + tan3 a 
r= B tang +7 tan? ¢ + Bx ) = See 
: -_ ps tans 9 : : 
In this expression —.— becomes sensible at the horizon ; 
3 3 . 
but © ar may always be neglected: wherefore, by substitut- 
ing the value of tan ¢, we shall finally get 
ia Bsin A 
a3 ae? 
Scos 2A49% +irh 2 
1 B2 sin3 A 
+ jou een, yah 3? 
{ cos? A+ 2i+ zt z 
1 63 sins A 
+5 -¢ 
fcos? A+2i +ir} 3 
At the horizon, sin A =1, and cos A =0; and, omitting in- 
sensible quantities, we obtain 
f B baa 1 pe 
If the values of 6 andi given by Laplace* he substituted in 
this expression, the horizontal refraction will come out 1290”, 
which agrees with the determination in the Mécanique Céleste, 
vol. iv. p. 246. 
The quantity of r will vary as 6 and 7 change with the state 
of the atmosphere. Now, £ is always proportional to the den- 
sity of the air: it varies, therefore, directly with the height of 
the mercury in the barometer, and inversely with the thermo- 
metrical changes. The standard height of the barometer being 
B, when the actual height is J, and the actual temperature ¢, the 
value of 6 will become 
B b 
1+mt ‘s B 
The other quantity 7 = - , varies with the thermometer only, 
* See Phil. Magazine for last September, p. 167. 
