422 Mr. Ivory on the Astronomical Refractions. 



equations. When this is done, the results are found to agree 

 very accurately with observations, the horizontal refraction 

 being no more than 10" above the French Table. 



But the foregoing hypothesis, although it agrees sufficiently 

 with the observed refractions, is opposed to Nature in another 

 respect. It limits the extent of the atmosphere above the 

 earth's surface to 25 miles, which is probably less than half 

 the real height. It therefore becomes necessary to take a 

 more enlarged view of the constitution of the atmosphere, 

 not confined to the exact proportionality of the altitude 

 ascended to the decrease of heat. This leads to the assump- 

 tion of the more general equations, 



m + l 



p- = (i-/)(f) " + St' 



££ = o-/)(^)"+/ 8 . 



The letters f and m are arbitrary numbers ; but they are 

 connected by the condition that the gradation of heat at the 

 earth's surface must agree with Nature. This is ensured by 



the formula f = P r ; 



And then m alone remains indeterminate. When ?w — 4, we 

 have the atmosphere already considered in which the heat de- 

 creases uniformly. When m is made greater than 4, we ob- 

 tain a series of atmospheres rising gradually above one 

 another in their total altitude, in each of which the rate of 

 the decrease of heat is not uniform, but continually decreas- 

 ing. The height through which the thermometer must be 

 carried at the top of the atmosphere, in order to depress the 

 thermometer one degree, is equal tofilx. (m-\-l); and it is 

 therefore very great when m is a great number. Yet all these 

 various atmospheres agree very nearly in giving the same 

 system of refractions ; the difference of the horizontal refrac- 

 tions, in the two extreme cases when m is equal to 4, and 

 when it is infinitely great, being no more than 17". The last 

 case, when m is a great number, is chosen as best suited to 

 the general circumstances of the terrestrial atmosphere. The 

 equations between the pressure and density are in this case, 



JL = 3 JL , ±.1_ 

 p *' e ' 4 e ' 



l±Bl _3.jl.J_ 



1+/3t' — . + 4 f ' J 



which equations are the foundation of my Table of Re- 

 fractions. 



In 



