(d) 



248 Mr. J. Ivory on the Constitution of the Atmosphere. 

 then, i + .r-_ca_ ^ . ^ ^ _ ^) _ 1±±I^1L ; 



and hence, observing that r' =.t — t, we get, 



If we suppose /3 = 0, the equations just found will be the 

 same as in the atmosphere of Dalton; and if /3 = 1, they will 

 belong to the atmosphere of equable temperature. When (3 

 has any mean value, the equation will determine an atmo- 

 sphere intermediate between the two extreme cases, and par- 

 ticipating of the nature of both. 



To find the numerical value of |3, we must employ the 

 equation (K). By the proper substitutions, we shall get, 



dx , I, 3/3 \ -\-aT 1 



(tt ( 1 — /3 I -\- ccT — al ) 



The inspection of this formula shows that -j— increases when 



/ increases ; that is, the decrease of heat becomes slower the 

 higher we ascend in the atmosphere. At the surface of the 



earth we have -^ = 90, and i = 0, and hence |3 = — = — 



very nearly. 



We may now compare the gradation of heat in this atmo- 

 sphere with the calculations already made. For this purpose 



put -^ = «, /= 1 - /3 = I-: then 



i^ == (1 - llY X (1 -fu) 



" ' = 1 — fn. Consequently 



log. -L -H log. 44-"-^ = 1+ 3 5 log. -L- ^ log. -^- \ , 



From these expressions we readily obtain this formula, which 

 is a sufficient approximation, viz. 



log. - - log. .— -^ = -r+ mrrn ' ^''^-T ' 



or in numbers, 



log. H log. — — = 5 -=- + -rrr log. . 



° P o 1 + « T 7 259 '' ;' 



In all accessible heights the last term on the right-hand side 

 of the formula is inconsiderable, and may be neglected; and 

 the calculations already made prove that the gradation of heat 



in 



