364 Astronomical and Nautical Collections. 



nearly the fluxions of fractional orders. Having first deduced from 

 the series which expresses the relation of the sides of any two 

 polygons, the general value of the product of two faculties (§.16) : 

 he transforms the faculty l'""", divided by m, into another which 

 he shows to agree in its general term with the series expressing 



the fluent °foo i"^^ ^ df, as it is obtained from the series of 

 Taylor, or of Bernoulli, for integration by parts. He derives also, 

 from a similar method of investigation, some very compendious 

 expressions for computing the same fluent for any other values of 

 tf and gives, at the end of his volume, some tables of their results, 

 which have lately been much extended by Bessel in his Funda- 

 menta. For the horizontal refraction, wliich is expressed by 

 V(» w*"^) (1 + ^'^ + ■B<+ . • .■) he finds A — .414214, 

 B = .262649, C = .200865, 'D = .160253, and £ = .132935 ; 

 (see Coll. No. XV.) : and from the values assigned by Laplace in 

 his Exposition, he computes the refraction equal to 7307 decimal 

 seconds at the freezing temperature, differing but little from the 

 7300 assigned to it by Laplace. Concluding, from observation, 

 that a uniform temperature of the atmosphere will not properly 

 represent the actual refractions, he suggests the alteration of the 

 quantity denoting the subtangent, or modulus of elasticity, in such 

 a manner as to correspond with the actual state of the phenomena; 

 and this is precisely what has since been attempted by Professor 

 Bessel. He also observes, that the refractions near the horizon 

 by no means follow the exact proportion of the densities, and gives 

 a table extending from 10° to 100° of Fahrenheit, which shows that 

 within these limit, the refraction varies in the ratio of 27 to 37, 

 while the densities are supposed to vary only in the proportion of 

 21 to 25 or 52 to 62. 



In the precise determination of the refractions very near the 

 horizon, Professor Kramp has not been particularly fortunate. The 

 terrestrial refraction, which is the subject of his fifth chapter, pre- 

 sents no particularly difficulty ; and neither of these investigations, 

 as belonging to the hypothesis of an equable temperature, presents 

 any remarkable interest at present. The method of Laplace, which 

 is well known from the M^canique Celeste, has deservedly su- 



