204 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



must find the numerical values of the quantities a and /. If ^ stands for the refraction 

 at 45° of altitude, determined very exactly from many astronomical observations^ we 

 shall have 



as will readily appear from the formula according to Cassini's method given in § 1. 

 MM. BioT and Arago have ascertained the value of a with great exactness in a dif- 

 ferent way, by means of experiments on the gases with the prism. In some of the 

 best attempts to determine a, the refractions at 45° of altitude, being reduced to the 

 barometer 29*6 and to the temperature 50° Fahr., are as follows : 



Dr. Brinkley 57*42 



De Lambre > . . . 57*58 



Bessel, Tab. Reg 57*55 



Experiments of MM. Biot and Arago .... 57*65 



Mean 57*55 



It appears that Bessel's determination has the best claim to be preferred : but as it 

 differs very little from De Lambre's result, which is adopted in the paper of 1823, 

 the same value will be retained in the calculations which follow. According to De 

 Lambre, the value of a is 60"-616* at the temperature 0° centigrade, and the barome- 

 tric pressure 0^*76 : wherefore, when the temperature is 50° Fahr. and the pressure 

 30 inches (= 0'"*762), we shall have 



762 1 



« = 60-616 X W X 1-Q0i8 X — ^ = ^^""^7 : 



and in parts of the radius, 



*a = -0002835. 



It has been found that L = 4347-8 fathoms at 0° centigrade or 32° Fahr. : where- 

 fore, if we make a = mean radius of the earth = 3481280 fathoms, we shall have at 

 the temperature of our climate, or 50° Fahr., 



4347-8 (l + —) 

 \ 483/ 



and hence 



LOjiM = ^^- _^-- = -0012958 ; 



a 3481280 ' 



X = -J = -21878. 



We can now inquire into the values of the last two terms of the foregoing formula 

 for the refraction, both of which are very small. With respect to the first of them, 

 we have, 



* Tableaux Chronomiques, publi^es par le Bureau des Longitudes de France. 



