250 On a new Calculation of the 



Tliese tables shew clearly the sources of the anomalies 

 to which I have adverted in the case of ^ UrsaB Majoris ; 

 the passages of the meridian below the pole give a latitude 

 obviously incorrect: compared with the result of all the 

 other series north of the zenith, it presents a difference in 

 defect amounting to more than 8'' for Montjouy, and to 

 nearly 7" for Barcelona; from which we may presume that 

 the zenith distances below the pole were nearly to the same 

 amount too great, and we may trace approximatively the 

 proportion of discordance which is due to each of the causes 

 which I have mentioned. ^ Ursa? Majoris was observed at 

 82|° from the zenith, an altitude for which the united endea- 

 vours of mathematicians and astronomers have failed to render 

 them master of the irregularities to which refraction is subject. 

 Tables have been formed on different hypotheses, founded on 

 the density and known temperature of the atmosphere ; these 

 tables have been compared with observation to verify the 

 theory, and to determine the constants for the most convenient 

 formulae. Other tables have been constructed, derived from ob- 

 servation alone, without regard to the physical explanation of 

 the phenomena ; and lastly, general formulae have been ad- 

 vanced for the refractions which, without reference to observa- 

 tion, are founded on the experiments of MM. Biot and 

 Arago on the refractive powers of the atmosphere, and on 

 those of Dalton and Gay Lussac, on the effects of changes of 

 temperature on its density. The agreement of observation 

 with all these tables so variously derived, from the zenith to 

 74° from it, shows that the refractions within such limits 

 are independent of the law of variation in the density of 

 the atmosphere. But beyond that limit, and more especially 

 beyond 80°, the calculation is no longer in accord with observa- 

 tion. The discordances vary from one day to another, and in 

 the same place, although the barometer, thermometer, and 

 hygrometer are stationary. It would be necessary to know the 

 differences of density and temperature of all the various strata 

 of the atmosphere traversed by the ray which passes little 

 above the horizon ; but meteorological instruments acquaint us 

 only with what is passing immediately around the observer : 

 thence the difficulty of representing low refractions by an exact 



