490 



Mr. Lubbock on the Heat of Vapours 

 a ( 1 -f- «) d a> 



d . 8 = sin x 



= sin 



*/cos 2 + 2ix + 3i*x* 

 a ( 1 + a) d co 



*/ cos 9 + 2 * «r 



[1] 

 _3 sin fl a (1 + a) e 2 t y 2 d co 

 2 



4- &c 

 (cos 2 + Zixf 

 [2] 

 The refraction will thus consist of two terms, which I proceed 

 to consider separately. The second term is minute, not amounting 

 to 2" sex. at the horizon. 



Throughout this treatise on Astronomical Refractions one accent 

 will be affixed to any symbol that it may denote the particular value 

 of the variable which obtains at the surface of the earth, and two 

 accents will be affixed when the particular value which obtains at 

 the superior limit of the atmosphere is intended. 



The limits of x 9 or x 1 and x" corresponding to u' and u", are 



= o, x" = u" 



, because «/ = 0, co" = 1 . 



xz=u co <=. u r fw; the function indicated by 



i i 



the letter f for the present may remain undefined. 



By Lagrange's theorem 



u = x + -- 1 x + TT^o 

 i 2r 



d (f x) c < 



= x + 

 Hence 



dx 



+ 



a 3 d 2 .(fff) : 



2 . 3 i 3 dx 2 



+ &c. 



CO. 



a d(f^r) 2 

 2x da; 



CO = f X + — : t - 



d 2 (f x) 



+ &c. 



2.3s 2 d x 2 

 Let x = *" - X, u = u" - U, co = 1 - F £7, then 





X= U- — FU 



i 



« « 2 d(F3Q« 



£7 = X + T F X+ ^ -jy- + &c. 



_ « d(FX)' 



« 2 d 2 (FX) 3 



2.3s 2 dX ! 



— &c. 



