ON THE MATHEMATICAL THEORY OF FLUIDS, 23o 



made for altitudes nearer the horizon, the differences between the 

 calculated and real refractions are of considerable amount, in 

 cases where the calculations have proceeded on suppositions re- 

 lative to the constitution of tlie atmosphere very remote from 

 the truth, and suffice to detect their inaccuracy. But it appears 

 fi-om Mr. Ivory's reasoning, that if p be assumed proportional 

 to p^, and the refractions be calculated accordingly, they come 

 out very nearly true quite close to the horizon. It would, however, 

 be wrong to conclude from this that the equation p — p^ repre- 

 sents the law of iiature, for the whole height of the atmosphere 

 calculated by this formula is found to be twenty-five miles, which 

 in all probability is far below the truth*. The fact is, astro- 

 noxnical refractions are very little influenced by the higher parts 

 of the atmosphere, so that supposititious atmospheres agreeing 

 with the existing atmosphere in the lower strata, and widely dif- 

 fering in the upper, may yet produce the same amount of refrac- 

 tion f. 



For the reasons given above no definite relation between the 

 pressure, density, and temperature of the air can be extracted 

 either from the observation of astronomical refractions or from 

 the theory of them. The only method that seems to be open 

 for increasing our knowledge of the constitution of the atmo- 

 sphere (and by consequence of elastic fluids in general) is to 

 multiply thermometrical observations at various heights and dif- 

 ferent stations, for the purpose of determining the law of the 

 mean distribution of temperature, and how far the variation 

 from one point to another depends on the variation of density 

 alone. Something in this respect may possibly be gathered from 

 the subject which next claims our attention. 



Theory of the Velocity of Sound. — The difference between 



* In place of the equation ^; = g'», Mr. Ivory assumes another, viz., 



/)= (1 — /) g " +/g2, /being an arbitrary quantity, which may have such 

 values assigned to it that the rate of decrease of temperature shall be slower as 

 the height increases, and the total height of the atmosphere be of any value 

 from twenty-five miles to infinity. This formula he employs in calculating 

 refractions, and finds them sufficiently accurate by taking/= ^ and n infinitely 

 great, which corresponds to an unlimited atmosphere, supposing the force of 

 gravity to be the same at all heights. 



t The memoir of M. Biot on astronomical refractions, read before the Paris 

 Academy, Sept. 5, 183G, and printed in the additions to the Connaissance des 

 Terns for 1839, treats the problem with all the generality and precision that 

 may be hoped for on a subject of this nature. 1 advert to the memoir here, 

 chiefly because its first part, on the conditions of the equilibrium of the atmo- 

 sphere, contains a lucid exposition of the mode of mathematically estimating 

 the effects of temperature, and of the mixture of aqueous vapour in the air. 



