164 On the new Method for 
° 1 : ee 
Again, the factor ie always between the limits 1 and 
ae! wherefore, if we put 
dr=BcosA X da ; 
J Sin? A + 2— — 26 
we may consider 7 as the exact refraction; for the true value of 
= 
ie 
which are so near one another that the difference will in no case 
amount to half a second. 
The differential expression of the refraction now contains only 
two variable quantities; namely, the height above the earth’s 
surface, and the decrease of the density of the air in ascending 
to that height. These two quantities are not entirely indepen- 
dent of one another. They are connected by a condition which 
depends on the pressure, and which we must now investigate. 
Let y denote the pressure of the atmosphere at the height x, 
measured by a barometer ; and 7’, the like pressure at the earth’s 
surface. Dr. Young supposes the pressure at the earth’s surface 
to be unit, and uses y to denote the relative pressure at any al- 
the refraction will be between the limits 7 and » quantities 
titude, equivalent to 4, when the symbols are taken in the sense 
here defined. If we suppose x to become x+da, y will become 
y—dy; and the small column of mercury dy will be equivalent 
in weight to the mass of air dx x x. According to Laplace the 
elastic force of air at the temperature of melting ice, whatever 
be the density, is measured by the weight of a homogeneous co- 
lumn equal in altitude to 7974 metres, or 4360°25 fathoms, 
At any other temperature ¢ reckoned on the centigrade scale ; 
: , 1 . 
and, allowing that air expands >— for every centesimal degree 
of rise of temperature; the length of the homogeneous column 
that measures the elastic force will be 4360°25 x (1-+'004 ¢) 
fathoms. Now, ¢ denoting the temperature at the earth’s sur- 
face, if we put /=4360-25 x (14-004 ¢), it is obvious that the 
column of mercury 7’ will be equal in weight to the column of 
air 7; for each measures the elastic force. Wherefore we shall 
have this proportion, 
yosdys: bx lydu:x 23 
dz 
whence, d. 7 =— 7 X% Finally, let S = > and i= 
7 then — = 7S; and, by substitution, we shall get these two 
a a 
equations which contain all the conditions of the problem, viz. 
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