MR. IVORY ON THE THEORY OP ASTRONOMICAL REFRACTIONS. 189 



earth ; because the pressures are proportional to the elasticities, we have the usual 

 equation, 



P _ 1 +/3r ^ 



or, which is the same, 



and by substituting the complete expression of the temperature as given in (5.), we 

 shall obtain, 



^ = c-»-/(c-«-c-^")-c-«X?)W: .... (6.) 



and if we omit the supplemental part, which is small even at great elevations, the 

 result will be. 



Now this is the constitution of the atmosphere in the paper of 1823 ; it is only ap- 

 proximate ; but it is an approximation applicable to every atmosphere that can be 

 imagined, requiring no more than that the quantity f have the proper experimental 

 value given to it. It is shown in the paper that the pressures and densities are thus 

 represented with no small degree of accuracy at the greatest heights that have been 

 reached ; which proves how slowly the supplemental part of the formula (5.) comes 

 into play. 



The foregoing manner of arriving at the constitution of the atmosphere adopted in 

 the paper of 1823, being drawn from properties immediately applicable to the problem 

 in hand, is more natural, and more likely to suggest itself, and more satisfactory 

 than the ingenious and far-fetched procedure of M. Biot, of transforming an alge- 

 braic formula for the express purpose of bringing out a given result. Laplace, 

 having remarked that the true horizontal refraction is contained between the like 

 quantities of two atmospheres, with densities decreasing in arithmetical and geome- 

 trical progression, conjectured that an atmosphere between the two limits, which 

 should likewise agree with observation at the horizon, would in all probability re- 

 present the mean refractions with considerable accuracy. It is upon this assumption 

 that the problem is solved in the Mec. Celeste, the observed horizontal refraction 

 being used for determining the arbitrary constant. Now in the paper of 1823 there 

 is no allusion to interpolating an atmosphere between two others ; a knowledge of 

 the horizontal refraction is not required ; the investigation is grounded upon a pro- 

 perty common to every atmosphere in a quiescent state ; and lastly, the resulting 

 table is essentially different from all the tables computed by other methods. If these 

 considerations be not sufficient to stamp an appropriate character upon the solution 

 of a problem, it would be difficult to find out what will be sufficient. But if it be 

 possible, with M, Biot's ingenuity, to trace some relation in respect of the algebraic 



