calculating the atmospherical Refraction. 165 
‘ dw (2) 
dr= cos A ee 
p A/Sin2A + 2iS—2Bw° 
Sida (i os tap F 
In these equations 7 has the same value with — in Dr. Young's 
Postscript. 
Every possible hypothesis relating to the density of the at- 
mosphere ; or, which is the same thing, every relation that can 
subsist between S and w, must be such, that the integral 
f—ds (1—w), taken between the limits » =0 and » = 1, must 
itself extend from 1 to zero. This condition being fulfilled, the 
second formula will determine the refractions in that constitu- 
tion of the atmosphere. . 
According to the method of Dr. Young, we must suppose 
w= Br+ Cr+ Dr? + Ert + &e. 
Now, if we write A for A Sin? A+2 - — 2Bw, we shall get 
: dw A : : 
from the last equations, —— = are by which the coefficient 
: ° Sin A 
B will be determined, viz. B = oan wv because 7, 5, are all 
evanescent together. Again, take the fluxions of the equation 
dw A i dda =) 1 $ ds ? da 
ir = pea} thus, Gat Tr BERR aw By eae 93 
i dw 1 1 
y but a 34 SW Bam AP wherefore, 
ddw i § CS) ?] 
“ia peat Cds ae at 8) 
ome 
z 
As we must suppose an equation between S and w from which 
the value of — will be found, the last formula will determine C, 
the second coefficient of the series. If we take the fluxions 
i ddS 
a B? cos2A iy dat 
the third coefficient D will be determined. And by continuing 
the like operations all the coefficients of the series may be found. 
We may also proceed in another way that will bring the deter- 
mination of the series more immediately, under the ordinary rules 
of analysis. For having an equation between S and w, we may 
4 dw da : 
again, we shall get, ——- x -5~3 by which 
SANG dS. 
thence find a value of S, and likewise one of zz» in terms of 
W5 
