12 Mr. Ivory 07i the Theory of the Astronomical Refractions. 



and, by substituting this value of u, the following formula 

 will be obtained : 



1 -| 



a 



This equation between the perpendicular elevation s, and the 



difference of temperature 



_ /3 K-t) • 



^- l+^r; ' 



contains the law according to which the heat decreases as the 



height above the earth's surface increases. 



Further, from the equation 



J, =(l-!7)c-^ 

 we deduce 



"'^'(^) = 



1 1 f f o 



^ ^ -u + log j3^ = w + 2'+ -^ +^ +&C. 



and, by substituting the value of q, 



6 ' * 



By means of this series and the value of o- in terms of u al- 

 ready found, it is easy to deduce 



and, by substituting the value of u, we finally obtain 



.= -^ = I„g|xf(l-l,-f^^.j^&c.)(B.) 



a 

 This formula determines a perpendicular ascent z, when the 

 difference of the pressures, and of the temperatures, at its 

 upper and lower extremities, have been found. 



The formulas that have been investigated are true in an 

 atmosphere of air mixed with aqueous vapour, as well as in 

 one of perfectly dry air; but in applying them, perspicuity 

 requires that the two cases be separately considered. 



[To be continued.] 



