Postscript to Dr. Young’s Letter on the Reduction of Experiments. P. 95. 
5. Corrections for Refraction. 
i. A simple and convenient method of calculating the precise magnitude of the atmospherical refraction, 
in the neighbourhood of the horizon, has generally been considered as almost unattainable ; and Dr. 
Brinkley has even been disposed to assert the “ impossibility of investigating an exact formula,” not- 
withstanding the “ striking specimens of mathematical skill, which,” as he justly observes, “ have been 
exhibited in the inquiry.” We shall find, however, that the principal difficulties may be evaded, if not 
overcome, by some very easy expedients. 
2 . The distance from the centre of the earth being represented by x, and the weight of the superincumbent 
column by y, the actual density may be called z, and the element of y will vary as the element of x and as 
the density conjointly; consequently dy — — mzdx; the constant quantity m being the reciprocal of the 
modulus of elasticity. The refractive density may be called i +pz, p being a very small fraction ; and it is 
easy to see that the perpendicular u, falling on the direction of the light, will always vary inversely as the 
refractive density, since that perpendicular continually represents the sines of the consecutive angles, belong- 
ing to each of the concentric surfaces, at which the refraction may be supposed to take place (Nat. Phil. II. 
s 
p. 8 1 :) and u z: ■ s being a constant quantity. The angular refraction at each point will obviously 
be directly as the elementary change of this perpendicular, and inversely as the distance v from the point of 
incidence; whence the fluxion of the refraction will be — = dr, as is already well known. 
v 
3. For the fluent of this expression, which cannot be directly integrated, we may obtain a converging 
series by means of the Taylorian theorem; but we must make the fluxion of the refraction constant, and 
. du d*u r z d 3 u r 3 
that of the density variable ; so that the equation will be u =, • r + ^ . — — H . — ■ + . . u being 
the initial value of u, when r ~ o. Now the whole variation, of which u is capable, while z decreases from 
s . d u 
1 to o, extends from ffp to s 5 or, since p is very small, from s — ps to s ; and dr being z: — , we have 
dv r z xdx—udu . du x dx 
the equation ps — vr + • — -f . . . But v — (x z — zr), du — - 
, and -j-— — . -7- — u ; 
dr u dr 
, , , . dy dr v dy 
and dx being — , and du — — psdz, — — - — . — . 
mz r dr mpsz dz 
4. We must now determine the value of the density z, which, when the temperature is uniform, becomes sim- 
ply zi y ; but for which we must find some other function ofy, including the variation of temperature ; and we 
may adopt, for this purpose, the hypothesis lately advanced by Professor Leslie, in the article Climate of the 
Encyclopaedia Britannica, and suppose the density to be augmented, by the effect of cold, in the proportion of 
1 to 1 +m | — — z j , n being somewhat less than -jL ; and since the density is as the pressure and the com- 
aarative specific gravity conjointly, we have z=y — z j > — z=i+~ — nz, d~ — 
%="- » d *. an d^l+4 + 2, consequently ^ = — 1 il + -22L + ”22.) 
yy zz d z z z z ^ dr mpsz\ z z 3 ' z ) 
zy 
, dv 
tnd -j- — 
dr mpszz 
+ 
nxyy 
+ 
nxyy 
— u. We may proceed to take the next fluxion with 
mpszz ' mpsz 4 1 r wicn re- 
pect to y, z, and v, the variations of u and * being comparatively inconsiderable: so that if we call 
- = X + Y + Z — s, its fluxion will be X 
r 
mt since 
dy — v 2 nvy 
1 dy 
2dz 1 
\ + Y 1 Zdy - 
2dz j 
\ ydr 
zdr i 
\ ydr 
zdr 1 
+ z 
ydr psz psz 3 
znvy dz —v , d z v 
- — , and -j- ~ — , we have — X 
psz zdr psz dr 
zdy 
4dz \ 
ydr 
«dr 1 
znvy 
+ z 
1 zv 
4 wy 
\nvy \ _ 
VX 1 
psz 3 
psz I + 
l/SZ 
psz 3 
psz )~ 
mp z s z \ 
zny 
psz 
zny 2 
z 3 
_znvy\ 
psz 3 psz ) + 
zn z y z z n z y 3 
