152 Royal Society. 



their being neglected ; and the equation is finally reduced to a form, 

 in which the remaining operations consist in investigating the inte- 

 grals of four expressions, and in subsequently assigning their nume- 

 rical values. Great skill is displayed in conducting these intricate 

 investigations ; and after going through the most laborious calcula- 

 tions and computations, the author exhibits a table of theoretical re- 

 fractions, deduced solely from the phenomena of the atmosphere, 

 for zenith distances, extending from 10° to 89|°. These refractions 

 are compared with those in Bessel's table, in the Tabulte Regiomon- 

 tancE, and also with those in the table in the Connaissance des Temps. 

 From this comparison, it appears, that the three tables agree within 

 less than 1", as far as 80° from the zenith: from 80° to 88° of zenith 

 distance, the numbers in the French table exceed those in Bessel's, 

 the excess being 2" at 84°, and 4" at 88° ; and with a single excep- 

 tion at 88°, (probably, judging from the character of the adjacent 

 number, arising from an error of computation,) the refractions in 

 the new table are nearer to Bessel's than those in the French table ; 

 but when the zenith distance is greater than 80°, the author con- 

 siders the accuracy of the French table questionable, both on account 

 of the hypothetical law of the densities, and because the quantity 

 assumed for the horizontal refraction is uncertain. 



After giving a few examples, illustrative of the use of the new 

 table, the author inquires how far the refractions are likely to be 

 affected by the term which it was found necessary to leave out, be- 

 cause the present state of our knowledge of the phenomena of the 

 atmosphere made it impossible to determine the coefficient by which 

 it is multiplied. For this purpose, the variable part of that term 

 has been computed for every half degree, from 85° to 88°, and the 

 results are exhibited in a table. From this it appears, that this co- 

 efficient, although considerably less than that of the preceding terra, 

 may still have some influence on the refractions at very low alti- 

 tudes. The mean refraction in Bessel's table, and in the new table, 

 can hardly be supposed to differ 2" from the true quantity, which 

 would Umit the coefficient in question to be less than one-tenth. It 

 is a matter of some importance to obtain a near value of this coeffi- 

 cient ; and it is probable that this can be accomplished in no other 

 way, than by searching out such values of the two coefficients as will 

 best represent many good observed refractions at altitudes less than 

 5°. If such values were found, our knowledge of the decrease of heat 

 in ascending in the atmosphere would be improved, and the measure- 

 ment of heights by the barometer would be made more perfect. 



At the end of the paper is given a table of mean refractions for 

 the temperature 50° Fahr. and barometric pressure 30 inches, at 

 every degree from 0° to 70° zenith distance, and at every 10' from 

 70° to the horizon ; and tables of the corrections requisite for va- 

 riations of the thermometer and barometer are subjoined. 



