6 Mr. Ivory on the Theory of the Astronomical Refractions. 



Now this is the constitution of the atmosphere in the paper 

 of 1823; it is only approximate; but it is an approximation 

 applicable to every atmosphere that can be imagined, re- 

 quiring no more than that the quantityy have the proper ex- 

 perimental 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 algebraic formula for the ex- 

 press purpose of bringing out a given result. Laplace, having 

 remarked that the true horizontal refraction is contained be- 

 tween the like quantities of two atmospheres, with densities 

 decreasing in arithmetical and geometrical progression, con- 

 jectured that an atmosphere between the two limits, which 

 should likewise agree with observation at the horizon, would 

 in all probability represent the mean refractions with consi- 

 derable accuracy. It is upon this assumption that the problem 

 is solved in the Mec. Celeste, the observed horizontal refrac- 

 tion 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 hori- 

 zontal refraction is not required ; the investigation is grounded 

 upon a property 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 con- 

 siderations 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 alge- 

 braic expressions, between the paper of 1823 and the calcu- 

 lations of Laplace, from which, after all, no just inference can 

 be drawn, it is not difficult to find between the same paper 

 and the view of the problem taken by the author of the Prin- 

 cipia, in 1696, an analogy much more simple and striking, 

 which deserves to be mentioned as it tends to bring back the 

 investigation to the right tract, which it seems to have left. 

 Newton, having solved the problem on the supposition that 

 the density of the air is produced solely by pressure, found 

 that the refractions thus obtained greatly exceeded the ob- 

 served quantities near the horizon : and hence he inferred, in 



