24 Further Remarks on Dr. Bradley’s Theorem 
it does not appear that any one of them at present will give the 
true refraction nearer than about six or seven seconds at ten 
degrees of altitude, and most of them reach to that extent with 
the same limits of error. It is therefore unnecessary to have 
recourse to the long and laboricus methods that have been of- 
fered, when more simple ones will effect the purpose with equal 
exactness. Our attention should rather be directed to the 
improvement of some one of the number, that is easily com- 
puted so as to extend its application and bring it to correspond 
better in observations made at low altitudes. But if it should be 
found that the law of its progression does not admit of being ex- 
pressed by a formula, we must endeavour to remedy the defect by 
means of a dalle, that wiil give its‘quantity at all necessary intervals. 
According to the theory published by Mr. Thomas Simpson, 
in his Mathematical Dissertations, aud further improved by Dr. 
Bradley, which, with modern determinations of the coefficients, is 
the one now mest commonly used in England, on account of its 
simplicity, the equation for the astronomic refraction is 
px tang.( Z.D—nr ) x ¢ + ) x ( a ) 
which is evidently of the indeterminate kind, since it contains 
no less than four unknown quantities, that require to be disco- 
vered from other sources. These are p, 7,7, and m3 of which 
the first, p, is the refraction at 45° of altitude of the object above 
the horizon, taken at any given standard of temperature and 
density of the atmosphere. The second, 7, is some multiple of 
the third, or mean refraction 7, by which the zenith distance of 
_ the object is to be diminished before its tangent is taken out of 
the tables. 
From a comparison of the theory with some accurate obser- 
vations lately taken, I have had reason to think that 7 is not 
a constant multiplier of 7, as has hitherto been supposed, but 
that it varies, according tosome function of the altitude of the 
object above the horizon. 
The fourth of these unknown coefficients, m, is the expansion 
of a volume of air, for cach degree of ascent of Fahrenheit’s 
thermometer ; and the same comparison abovementioned, in eon- 
junction with the latest and most exact experiments of two justly 
celebrated modern chemists has also furnished some strong rea- 
sons for doubting, whether the expansion for each degree of the 
thermometer is the same for all states of temperature, from the 
freezing to the boiling point. 
This inquiry may, perhaps, by some persons be deemed in- 
teresting, as it points out a subject where the determinations 
of the chemist are corroborated by those of the astronomer. 
These 
