376 Asironomical and Nautical Collections. 



which agrees so nearly with the results of Laplace's hypothesis, 



in which ~ = 1.396 (Coll. XIV.) 



There is a mathematical paradox in the latter part of Mr. Ivory's 

 paper, which requires some further explanation. 



" The density in the hypothesis of Kramp" being too compli- 

 cated for calculation, he deduces from it, says Mr. Ivory, " this 

 more simple value," z = e"'""' " by retaining only the part of 

 the expansion of the function in the index that contains the first 

 power of cr." 



" In all this Kramp is followed by Bessel, whose aim it is to 

 determine the value of e that will best represent all the observa- 

 tions of Dr. Bradley, without paying any regard to the terrestrial 

 phenomena, or to any further theoretical considerations whatever. 



" Now, there is an essential distinction between the rigorous 

 expression of the density, and the approximate value used instead 

 of it. The latter belongs to a finite atmosphere, and the former to 

 one of unlimited extent ; . . . and the total height will be determined 

 by the equation" e"""*^'— t = 0. 



" At the surface of the earth, we ought to have s = — ; which 



" 5 



would limit the atmosphere to about double the height in the hypo- 

 thesis of Cassini. Bessel determines e =: — nearly : which is 



28 ' 



quite inconsistent with the value of" ^ " at the surface of the 

 earth, and with the elevation necessary for depressing the ther- 

 mometer one degree, as found by experiment. Accordingly, 

 although the refractions in this table represent Dr. Bradley's obser- 

 vations with great exactness as far as 86° from the zenith ; yet, at 

 lower altitudes, they diverge greatly from the truth." 



Now it seems obvious, that since z = e~^'~'^', when e~^'-'>— e 

 = 0, z — £ ~ 0, and z = i, instead of z =: 0. However this 

 may be, Professer Bessel certainly states the result very differently 

 from Mr. Ivory, for he says, (Fund. Astr. p. 57,) 



" Finem huic capiti imponat atricta densitatis aeris comparatio. 



