28 Further Remarks on Dr. Bradley’s Theorem 
or, t,(ZD—nr) x (l— mé), 
and substitute for 2 and for @ their two extreme values taken from 
the foregoing table of means, we shall have in one instance, 
t, (ZD—3:74r) x (l—m x 32), 
and in the other, 
t, (ZD—3:47r) x (1—m x 56), supposing the 
formula made for temperature zero, 
Now in these two equations, the lower the mercury is in the 
thermometer, the greater is the value of 2 found to be: and 
the higher it is on the contrary, the less is the value of 2: but 
to obtain the two values of 2 equal in the two equations, it is 
evident that the multiplier m must be increased at 32, and di- 
minished at 56; which seems to be a further and interesting 
corroboration, of what was before stated, viz. that the rate of 
expansion for the low temperatures is greater than that for the 
high ones, : 
By carrying this mode of reasoning a little further, and sup- 
posing in the two last equations, that the values of ¢, (Z2D—mr) 
are constant, it would in that case be evident, that the variations 
of m would be inversely as the numbers 32 and 563 or that 
the rate of expansion would be in the inverse proportion of the 
height of the thermometer, provided the refraction and the rest 
of the coefficients remained the same. But as this is not the 
case, and the result thus obtained is far greater than is proved 
by experiments, and also that the ratio of the two values of m, 
in the two equations, is considerably nearer a ratio of equality, 
it follows that the value of n must also vary, and possibly, ac- 
cording to some function of ZD, or the zenith distance of the 
object. 
The difficulty of determining the values of m, under all tem- 
peratures of the atmosphere, and of at all altitudes, above the 
horizon will, from this, become. evident: and it shows that 
they can only be obtained by some such method as that of the 
least squares proposed by M. Lagrange*, and from a long series 
of observations taken at all periods of the year, in order to find 
them for the whole extent of the ranges of the barometer and 
thermometer. 
Were all other circumstances to remain the same, and only 
the value of m to change, we might easily obtain it in the fol- 
lowing manner : 
The mean refraction r is to the apparent refraction @ as I to 
~- poorrzgril +m6:1; and therefore r=e(1 +b), 
consequently m= oe = ae and by taking the different 
* New Method of determining the Orbits of Comets. Paris, 1806. 4to. 
values 
