113 



At 87°. 42'. 10", refraction = 59", 5 tan. (87'. 42^. tO" — 

 3,803 r)* =16'. 51", 0. 



This refraction is less by 35",5 than the mean of the ob- 

 served refractions. Hence we may safely conclude that the 

 actual quantity of refraction is between the results from an 

 uniform temperature and from a density decreasing uni- 

 formly. 



Laplace has shewn the same from the horizontal refrac- 

 tions computed on each hypothesis, and compared with the 

 observed horizontal refraction. But it does not appear that 

 the mean observed horizontal refraction has hitherto been as- 

 certained with much accuracy. 



Laplace has also in the case of uniform temperature in- 

 tegrated the fluxional equation for refraction, in which he 



* This form or r=zk tan. (S — nr) may be readily computed by help of an auxiliary 

 angle y. 



log. tan. y = log. tan. fl -f- log. ( i^„fc^„. i„ ) + i log- 

 then log. r = t log. ^ - ^^ ^„ + log. tan. \ y 



_ , , . . fc t*.n. 8 — ft tan. n r 



For h tan. (9 — nr) = -— j— — — 



^ ' l+tan.itan.nr 



Hence -L-^— ;^— = tan. 9 



n«tn. 1'' 



let taa. iyzzr V y- sin. 1" 



then 'Ji!lJl^!L2!L V -~t— tan. y = tan. 9 

 a ft *«'«• 1 



Whence log. tan. y =: tan. 9 4- &c. 



&c. &c. 



VOL. XII. R 



