MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 209 



sating the small changes which a difference of temperature causes in the mean con- 

 stants. 



Experience confirms what has been said ; for all the astronomers who have at- 

 tended to aqueous vapour in the atmosphere, agree in admitting that it either has no 

 influence, or but a very small and imperceptible effect, to alter the refractions. On 

 this head it will be sufficient to cite the authority of M. Biot*, who seems carefully 

 to have studied this point, on which he expresses himself very strongly. The very 

 exact coincidence of the theoretical with the observed refractions as far as 88° or 

 88j° from the zenith, concurs to prove that the variable quantity of vapour in the air 

 has little influence so long as it retains the gaseous form ; but at lower altitudes, 

 when the rays of light become almost parallel to the horizon, it is very probable that 

 particular and local causes may come into play. 



1 1 . Nothing is now wanted for completing the solution of the problem, except the 

 reducing of the expression (C.) to a form fit for numerical calculation. 



Investigation of the integral Q^. 



We have 



assume. 



Q„ =/"£--=/ 



edxc~ '^ 



01_^)s + 4.^.^ 



= /y/(l -e2)2 4_4e2.^=l -e2 + 2e2sj; 



then, 



e dx , 



— ^ =z € .maZf 



— = t=l — e'^(z- «2), 



By Lagrange's theorem, 



'¥ = t-t% 



* = ^ + ^'-^ + 172 --^ + 17273 •-TF- + ^^-* 



mdz=^mdt. '^l + e'^'-^ + TT^ ' -Ji^ + TTTs "dW + &c. j ; 

 consequently. 



Wherefore, if we assume 



Qo = Ai e + A3 e3 + A5 e5 + &c., 



* Precis Elem. de Physique, p. 229. torn. ii. edit. 2nd. Addit. k la Conn, des Temps, 1839, p. 36. 

 MDCCCXXXVIII. 2 E 



