Pierre Simon Laplace.

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different heights.* In this way, Bouguer found that the light of a body, seen
in the zenith,, is reduced, after having passed through the atmosphere, to

[8596] 0,8123. The tabular logarithm of this number is log.^ == —0,0902835 ;

for'tho''' therefore, by dividing this logarithm by the sine of the apparent altitude of the
heavenly body, we shall obtain the logarithm, of the intensity of the light at
that altitude.

Very near the horizon, the diminution of light depends, like the refraction,
upon the constitution of the atmosphere. If we adopt the hypothesis which
we have given in [8411, 8412], we shall easily obtain, by the analysis of
that article, the corresponding value of the intensity of the light. But we
may, without fear of any sensible error, use the hypothesis of a uniform

[8598] temperati^re. In this hypothesis, we havef pdr = — Idp; therefore, by

of the
light o
beavei
bodies

[8595]

intensity.

[8597]

[8593i ] atmosphere at the zenith of that place, or to s= c , if the distance from the

zenith of that place of the sun's body, or the inclination to the vertical, be represented hy
0. We may finally remark, that we may use either tabular or hyperbolic logarithms

[8593n] jp, [8593], because they are proportional to each other, and occur in both members of the
equation ; and the same may be done in the expression of log.E [8592'] ; taking care,

[8593o] liowever, to adapt the constant quantity Q to the kind of logarithms which are used;
being the tabular logarithms in [8596, &;c.].

* (3970) At the zenith, where cos.0=l, the observed intensity of the light is E,
r8595al

[8585, 8592] ; the general value, corresponding to any zenith distance ©, being s,

[8585,8584]. Now if the ratio of tliese intensities is observed, and found to be as 1-to

[8595&] j^ ^,g gj^^ii Y\^\Q s=bE, or log.s = log.6-f log.i^; substituting this in [8593], we

■^ loCT.JS , , _, COS.0 , 7 , -1 1 J

[8595c] get log.6-j-log.£= ; whence log.jE= . Iog.6; whence we easjiy deduce

the value of E, as in [8596].

t (3971) The differential i
[8559a]; multiplying this by — Zp, we get — ldp = p.dr, as in [8598].

t (3971) The differential of the logarithm of [8299], is -l = _ ^ = f nearly,

[6598a] ' V ^ ^ ^ ■' p I I

X, iii. § 12.] HEAVENLY BODIES IN THE ATMOSPHERE. 555

putting the element of the refraction equal to d^, we shall have very
nearly,*

[8599]

Bion of the

H being a constant quantity. Therefore the logarithm of the intensity '"tensity,
of light, of any heavenly body, is proportional to its refraction, divided by the

m terms
of tho

[8601]

cosine of its apparent altitude [8599^]. refraction.

We have seen in [8503], that, at the apparent altitude of 50°, the
refraction is 186",728; and in the hypothesis of a uniform temperature,
the refraction in the horizon is 7390",71 [8370] ; hence we easily find that

the light at the horizon isf . We may, by these formulas, determine, [8602]

in an eclipse of the moon, the quantity of light which falls upon the moon's

* (3972) If we neglect — -/pjf, '^•(0' in comparison with 1, and put - =;= 1, [8599a]

as we have done in the preceding notes, we shall find that the radical, in the denominator

of the value dd [8262], becomes y/l— sin.9 e = cos.© ; consequently this expression of rgKogi-i

dd 3 reduced to <?a = — — - . dp. — - = —- . 5-^ . [8598] ; but from [8586A], we

n2 ' CO3.0 n2 I COS.0 *- -" "■ -^ f 8599c 1

pdr ds 1 , , ... ,. ^K 1 ds . . .

have ■ — = — • TT ; hence by substitution dd = — — r • tt, • — .sin.e : and by

COS.© £ Q n^ Ql s

putting - • ; == — , it becomes dd = — 77.— .sin.e: multiplying this by -: — . we
^ ° n^.Ql H He ' JT ./ o J sin.©'

get [8599]. Its integral is log.s = — H.- — ; 55 being the whole astronomical refraction. [8599rf]

This result is the same as in [8600].

I (3973) If we put / for the value of e, when the zenith distance is © = 50°, and

186" 728 [8602a]

the refraction 53 == 186",728 [8601], we shall have log.s' = — if. — -^ [8599^]. At

the horizon, where = 100°, and the refraction 5^ =7390",71 [8601], we shall suppose
that s becomes s", and we shall have log.s"=— jff.7390",71 [8599(^J. Dividing the [^6026]
expression of log.s", by that of log.s', we get,
log.g" 7390",71.v/ii

log.s' 186,728

= 39,5801./!; or log.s"=: 39,5801. v/l-log./. [8602c]

\og.E
Now when © = 50°, we have, from [8593], log.s'= — — -; hence,

log.s" = 39,5801 X log.i: = — 3,57343, [8602rf]

using the value of log.-E [8596]. The natural number corresponding to this logarithm
is ^j-j ; hence we have s" = ^j^, which differs a little from the result in [8602J.

556

EXTINCTION OF LIGHT

[Mec. C6I.

[8602'1

[8603]
[8604]

Sun's light
most in-
tense near
the centre
of its disk.

[8605]

[8605']
[8606]

[8606']

disk, in consequence of the refraction of the sun's rays in passing through
the earth's atmosphere, taking also into consideration the extinction of the
light during that passage.

13. According to the experiments of Bouguer, the light of the sun'^s disk
is less intense, near its limb, than at its centre. At a distance from the limb
equal to a quarter of the semi-diameter, he found the intensity of light to
be less than at its centre in the ratio of 35 to 48. Now any portion of the
sun's disk, when it is transported by the sun's rotatory motion from the
centre towards the limb of its disk, ought to appear with a more brilliant
light, since it is viewed under a less angle ; and it is natural to suppose that
each point of the sun^s surface emits an equal quantity of light in every
direction. We shall put e for the arc of a great circle of the sun^s surface,
included between the luminous point and the centre of the sun^s disk, the
sunh radius being taken for unity ; a very small portion a of the surface,
being transported to the distance o from the centre of the disk, will appear
to be reduced to the space* a.cos.e ; the intensity of its light must therefore
be increased, in the ratio of unity to cos.o. But on the contrary it appears
to be diminished. This difference is easily accounted for, by means of the
atmosphere surrounding the sun. We have seen in the preceding article

[86O60]
[86066]

[8606c]
[8606rf]

[8606e]
[8606/]

[8606^]

* (3974) We shall suppose, in the annexed figure 111, that F is
the sun's centre ; FAB the line drawn towards the observer on the
earth; AC the sun's surface; BDE the surface of the sun's
atmosphere; AA' = CC an infinitely small arc of the sun's surface ;
the line 1/C7 is perpendicular to FAB; and the lines A'B', CD,
C'W, parallel to AB ; lastly, the angle CFA = 0. Then if a part
of the sun's disk, whose base is AA', height above the plane of the
figure A, and area hxAA' = (i, be transferred, by the sun's rotatory
motion about its axis, which is supposed to be perpendicular to the
plane of the present figure, until the arc AA' arrive to the situation CC, the base CC ,
when viewed from the earth, will appear under a less angle than when in the situation AA
in the ratio of cos.e to 1 ; so that the apparent magnitude of the base CC is reduced to
CI=AA'.cos.Q. IVTultiplying this by the height A, which is not altered by the rotatory
motion, we get A X«^«^'Xcos.0 = a.cos.e [8606c], for the reduced value of the part a.
Now the part CC sends forth as many rays in a direction CD, parallel to AB, as the
part AA' does in the direction parallel to AB. Therefore the intensity of the light at C,
to that at A, must be as AA' to CI, or as CC to CI; that is, as 1 to cos.e, as
in [8606'J.

X. iii. <§. 13.J IN THE SUN'S ATMOSPHERE. 557

/_

COS.0

that the intensity of light which results from it is equal to* c , c being [8607]

the number whose hyperbolic logarithm is unity. Now the intensity of

light is c~^ at the centre of the disk ; therefore at a point which is distant [S<^08]

f_

from the limb by a quarter of the semi-diameter, it will be .c

COS.0

[8607/] ; sin.© being equal to | ; therefore we shall have,!

I6 -^-V/^

35 -/

— .0

— — c

7

48

[8609]

[8510]

This equation gives the following value of f;

/= 1,42459; [86ii]

* (3975) We have seen in [859Sk, I], that a ray of light, whose intensity is represented
by unity at the surface of the sun's body, will be decreased to c""-^, in passing vertically
through the sun's atmosphere, in the direction AB or CE, in fig. Ill of the preceding [8607i]
note. Moreover, if the ray pass through the sun's atmosphere in the oblique direction CD,
forming the angle ECD = ©, with the vertical FCE, its intensity, upon quitting the

/

[8607c ]

sun's atmosphere at Z>, will be represented by c [8593Z]. This intensity at the [8607rf]

point D is to be increased in the ratio of 1 to cos.©, as in [8606'], because it is
supposed that as many particles of light proceed from the surface CC, between the
parallel lines CD, CD', as in a vertical direction, or parallel to CE. Hence the

f_

X COS f^

intensity at the point D becomes .c , as in [8609].

[8607e]
[8607/]

[8609a]

f (3976) If the point C of the sun's disc, figure 111, page 556, be supposed to be
distant from the sun's limb by ^ of the semi-diameter, we shall evidently have
sin.CjFl4 = sin.© = 1; whence cos.©=y//^. Substituting these in the expression of [86095]

_ — /V¥
the intensity at the point I> [8607/j, it becomes \/y^-.c ', while at the centre of [8609c]

the disk, at B, it is represented by c"^ [8593A:]. Now, according to the observations of

Bouguer, these quantities are to each other as 35 to 48 [8604] ; hence we easily obtain

the equation [8610] ; or, as it may be written,

[8609rf]

•^•(^"77) 35 v/7 hyp- log. J. -^
c = — . — ; whence /= 5 = 1,42459.. ; [8609e]

^""77

as in [8611]. This value of / represents the hyperbolic logarithm of c^; whence we rQp^n-,
get c-f [8612], nearly.

VOL. IV. 140

568 EXTINCTION OF LIGHT [Mec. Cel.

whence we obtain,
[8612] c-f == 0,240686 = E, [8693c]

From this it follows that the intensity, at the centre of the sun'^s disk, is
^®^^^^ reduced, by the extinction of the light in the sunh atmosphere, from 1 to

0,240686. A column of air at the temperature zero, and under a pressure
[8614] corresponding to the height of the barometer 0'"'*'%76, must have a height

of* 54622'"'*'"', to decrease the light in this manner. This will therefore

be the height of the sun's atmosphere, reduced to the preceding density, if,
^^^^^ ^ with the same density, it extinguishes the light as in our atmosphere.
Thebrii- Hence ice see that the sun would appear much more luminous, if the
IhTsunIs atmosphere which surrounds it were taken away. To determine how much
no8?her;. its light is weakened, we shall observe that, by supposing the sunh
[8615] semi-diameter equal to unity, and putting cos.e = x, the whole light will

[8616] bet ^'^f^dx.c ^. It is true that the intensity of the light is sensibly

* (3977) Substituting the value of log.J5: [8596], and that of I [8275], in [8592'J,
[8614a] after dividing it by —I, we get Q.(p)= '074"! == ^^^^^ • Now upon the hypothesis

[86146]

7974™ 88322"

assumed in [8614'], Q.(p) is the same for the sun as for the earth ; therefore this
value of Q.(p) will correspond to the sun's atmosphere ; and by substituting it in the
equation [8593a], we get Z=— 88322«'^"«Mog.£; and since log. ^ = —0,6185492,
[8614c] [8612], we get Z=88322"«t^«» X 0,6185492; being nearly the same as in [8614],
which represents the height of the sun's homogeneous atmosphere, supposing it to be of a
uniform density, as in [8614].

t (3978) We shall suppose the arc CC, figure 111, page 556, to revolve about the

[bblbaj Yyi^Q FAB, as an axis, so as to describe, by its revolution, an annulus, whose surface is

[86166] 2*. CjH. CC; 2ff being the circumference of the circle AC, whose radius is unity,

[8605']. Now AC = e [8605]; hence CH — s'm.e, CC' = de; therefore the

expression of the surface of this annulus is,

[8616cri 2*. CH. CC = 2<ff.sin.0.<Z0 = — 2*.6^.cos.e = — ^tt.dx, [8615]

To obtain the intensity of the rays which proceed from this annulus, in the direction
parallel to FAB, we must multiply the surface of the annulus — ^if.dx [8616cZ], by the

f_ _/

[8616e] expression of the intensity [8593Z], c '^°^'^, or c "^ [8615]; and it becomes
_ /
— 2ir.cZa;. c "" . Its integral between the limits x = \ and cc =0, gives the whole
•■ •' ■' intensity of the sun, supposing it to be covered with an atmosphere ; hence this intensity

X.iii.<§>13.] IN THE SUN'S ATMOSPHERE. 559

proportional to c , only between the limits = and = 88° ; and rggj^n

beyond this last limit the intensity follows another law [8616n, 0, j?]. But

the sine of 88° differs so little from unity, that we may neglect the portion [8617']

of the solar disk which corresponds to this difference, or else assume, as in

^ ' [8618]

the other parts of the disk, that the intensity of light is proportional to

_/

c . Therefore if we suppose that the intensity of the sun*s light is [8619]
represented by unity, when its atmosphere is taken away, or f equal to

nothing [8616A], we shall have f^dx.c '' for the expression of the [86i9'j
intensity, in its decreased state, by the action of the sun's atmosphere.

To obtain this integral, we shall put — = o, and z = — ; and then it [8620]

.J 9^

— L _I

is expressed by — '^it.f^dx.c *; or, as it may be written, ^it.f^dx.c "^ , by merely [8616^-]

changing the order of the limits of the integral. If we suppose /=0, as in [8619'], the

/ [8616A]

intensity of any ray c ' [8616e], becomes equal to unity, being the same as when

[8616i ]
the atmosphere is taken away [8593Z:] ; and in this case the expression [8616^] is reduced

to the form 2it.f^dx = '^. Hence it appears, that if the sun's atmosphere be taken [8616&]

away, the whole intensity of the light, proceeding from the sun's body, will be represented

by 2w; but if it have an atmosphere, the whole intensity will be ^'^.f^^d^.c "^ [8616g-]; [8616/]

so that if the whole intensity of the sun's light, undiminished by the atmosphere, be

[8616m]
represented by unity, its actual intensity, when diminished by the action of its atmosphere,

_/

will be represented by f^dx.c * , as in [SGIO'J. These results require some modification, [8616n]

in consequence of the terms which are neglected in [8589, &lc.] ; since these terms impair

[86160]

the accuracy of the formula [8609], when © becomes large; in the same manner as we

have seen, in [8483], that the formula for the refraction [8474] cannot be used when rgeieni

exceeds 88°, on account of the neglect of similar quantities in computing that formula.

The effect of these neglected terms, in computing the intensity of the sun's light, are not rggjg •,

however of much importance, because sin.© is nearly equal to unity, as is observed in

[8617'J.

560 EXTINCTION OF LIGHT [Mec. Cel.

— g- ; the limits of the integral being from 2; = co to

'^V

[8622]

/dz,c-^ c"" ( 1 1-2 , 1.2.3 1.2.3.4 , „ > ,

ro^ooi The integral must be taken from 2:=co to 2 = — = /'; so that the

constant quantity is nothing ; consequently the integral becomes,

_ /
5 - ^==^.c - ^.|l-1.2.^+1.2.3./-1.2.3.4.93+&c.}=/o'£/a;.c . [86i9>c]

We can reduce this series to a continued fraction, by the method explained
in [8340, &c.]. For this purpose we shall put,
[8625] u = 1 —2q.{\—t) + 1.2.3.9^.(1— 0'—&C. ;

and we shall have,t

* (3979) If we change, as in [8620], the constant quantity / into — , and the
[8620a] 9

variable quantity x into — , we shall get — =z] substituting these in [8619'], it

[86206]

becomes as in [8621]. The limits x=0, x = l [8619'], being substituted in z [8620a],
give the corresponding limits of 2, as in [8621]. Now we have generally,

as is easily proved by taking the differential and reducing ; therefore if we put successively
'■ ■' m = 2, m = 3, m = 4, &,c., we shall get, by repeated substitutions,

*dz.cr'
IT

-f'-^ = ^. +2./-^ = ^ -2. ^ -2.3. f

t/ qz^ qz'^ O qz^ qz'^ qz^ */ qz

= — , -2. —3 + 2. 3. '— + 2.3.4. n^ ; &c.

qz^ qz^ qz^ v qz^

This last expression is easily reduced to the form [8622] ; which vanishes at the first

[8620/] 1

limit 2^ = 00 [8623] ; and at the second limit « = — [8623], it becomes as in [8624].

t (3980) IVIultiplying [8625] by (1— 0^ we get,
[8626a] u.{\—tf = (l_^)2_2^.(l—<)3 + 2.3.^2,(1— O^—^c.

Its differential, being multiplied by ~ , considering w, t, as the variable quantities, gives,

[8628]

X.iii.<^13.] IN THE SUN'S ATMOSPHERE. 561

nil

q. ^ ,(l—ty—2qu.(] —t)—u + 1=0. [8626]

We shall consider u as the generating function of i/^, so that we shall have,

u = yi + y2't + ys.t^ +2/,^i.r + &c. [8627]

Substituting this value of u in [8626], and then putting the coefficient of
r~^ equal to nothing, we get the following equation of finite differences ;*

qr, y^+, — (2gr+ 1 ). y. + gr. y,_, = ; [8629]

in the case of r = 1, this coefficient will give,

0=:qy,-{2q + l).y, + l; [8630]

which may be included in the preceding equation, by supposing yQ=z — . [8631]

Now the equation of finite differences in y^ [8629], gives,!

f-' 2yr+ 1 y,+i

f qr }f

[8632]

q. ^ . {i—tf—^qu.{\—t) = —2^.(1— 0+2.3.92.(1—^)2— 2.3.4.^3.(1— 0^+&c. [86266]

Adding 1 to both members of this equation, and then substituting for the second member

its value M [8625], we get q.^.{\—{f—'ilqu.{\—t)-\-\ = u, as in [8626]. This is

'^^ [8626c ]

easily reduced to the form [8626</], by observing that ~77~ =^ "^I • ^ "l~ "' ^"'^

dt dt

du ^ d.{ut) , d.{ut^)

* (3981) Substituting the value of u [8627] in the first four terms of the equation
[8626<Z], and retaining only the quantities depending on i^~^, we get the four following
terms respectively ;

{qr.yr+i—'iqr.yr-\-qr.yr-i — yr]'i''~^' [8629a]

Now to satisfy the equation [8626] for all values of t, the coefficient of t*'~^ must be put
equal to nothing ; hence we get [8629], by a slight transposition of the terms. The rggggj-i
coefficient which is independent of /, being also put equal to nothing, gives [8630].

t (3982) Dividing [8629] by or.y^, and transposing the two first terms, it becomes
as in [8632]. Now if we multiply the assumed equation [8633], by

{\-\-q.\r — 1]+2:,.). ?zzJ , it becomes l+g'-[r — 1] +Zr= g-r. i^ ; and by substituting

y^ V'- [8632b]

the value of ?^ [8632], we get l+a.(r— 1)+^; =2g'r+l — or. ?^; or by reduction

yr ^ ' ^ yr '

VOL. IV. ^ 141

562 EXTINCTION OF LIGHT [M^c. Cel.

We shall now suppose,
[8633J y*- — ?'*

and we shall have,

q^.r.(r-{-l) ^

z.

[8634, ,, = g.(.+l)_Xl_i_J_,

[8635] z, =

'r+l

q-{r-\-\)

1-1-

qr

Hence we deduce.

1 + ^^

[8636] ' , 2y

1 -f-

1 +

4g

1 +

3?

therefore,*

1+&£C.

[8637]

y. 9

?

yo i+^x

1-

,3,

^ Stq

[8632c] Zr=^ q.{r-\-\\ — qrJj^. If we substitute, in this, the value ^!:±1 = — g-(^+l) ^

yr yr l+^r+2r^i

[8632(fl deduced from [8633], by writing r+l for r, it will become g^==g.(r-f !)■— ^ ^-(^+1)

l+^r+

-«,

[8632e] ^, = c.(r+l). \ 1— ^I \ = g.(r-}-l). ,^+^'-+^ = ?.(r-|-l) L

- r+l

as in [8634]. This may also be put under the form,

1+

as in [8635]. From this we get, by putting successively r=l, r==2, r = 3, &c.,

[8632/1 ^-= % ' ^~rrir' ^^ttw: '

■^1+^ "^1+^3 "^1+^.

and by successive substitutions we obtain z^, as in [8636].

* (3983) Putting r= 1 in [8633], we get ^^ = — i— ; and by substituting ar,,
[8637a] yo 1+^1

[8636], it becomes as in [8637].

X. iii. § 13.] IN THE SUN'S ATMOSPHERE. 663

Putting the two expressions of u [8625, 8627] equal to each other, we get,*

y, = 1—1.2.^+ 1.2.3.9^—1.2.3.4.9-^ + &c. t8638]

Moreover we have y^ = — [8631] ; hence we obtain,!

* (3984) The part of u [8625], which is independent of t, is evidently equal to
l—Qq-^-l.^.S.q^—hc; and the corresponding part of u [8627], is y,. Putting these [8638a]
two expressions equal to each other, we get [8638].

1 . _ q

t (3985) Substituting y^ = - [8631], in [8637], we get qy^ =

^ 1 _|_ % [8639o]

1 + i .

1+&C. 5

multiplying this by cr^, and then substituting the value of y^ [8638], we obtain [8639].

_ /

The first member of this expression is the same as the value of f^^dx.c ' [8624] ; and

by using the symbols [8641], supposing also the fraction [8645] to be represented for

brevity by F, we shall have,

f^'dx.c ' =q.c-f.F= s^i)

1 + ^ V [86396]

] + !

1 +

1+&C.

The continued fraction F, arranged as in the second member of [8645], is similar to that
in [8362a]; changing the numerators q, 'Hq, Sq, &c. into e^^ s^^\ s^ \ &c. respectively ; [8639c]
also i into r. Hence we obtain, as in [86326], the following series of fractions for
determining the value of F;

5 &c. [8639d]

I &c. [8639e ]

i+icT^i^r^; «"=• [8639/]

Value of r+1.

1

2

3

4

Upper index.

1

1

1

1

1

0'

1 '

1
1 '

1

1+SV2)

l+£(^)'

(l-|_£Cl))_|_sC2) '

Abridged forms
of the fractions.

1

'

1 '

J) (.3) '

Lower index.

1

s(0.

s^2)5

gw.

8<«;

Hence iV^

1^ = 1

J

JVC2) =

1;

iV(3)==:l+s(2) =

&c. [8639g-]
s^« ; &c. [8639^]

as in the general formula [8643]. In like manner,

jD") = 1 ; D^^^ = 1+s^" ; D(3) = (i_|_s(i))-|-s(2) = D(2)-|-s(2\D

X>(4) ^ (l-j-sCi)+s«>)-|-e(3\(l+£f") = D^^+P\D^^, &c. ;
as in [8644].

(1)

[8639i ]

[8639*]

564 EXTINCTION OF LIGHT [Mec.Ca

[8642]

[8643]

if ita at
mosphi
were r
moved

[8646a]
[86466 ]

9.c-/(l~1.2.g+1.2.3.^-&c.) == -^ ^f.'dxx

f

[8639] 1 -f- ?^

1 +

1 + ?^

1+'^

l+Sic.

[8640] We shall put,

[8641] s(^)=:2^; E(2> = g; s(3) = 3g; s(4) = 2^; s(«) = 4^; s<«) = 3y; 6('> = 5g, &c.;

and shall then form a series of fractions, beginning with — and .

^ ^ 1 l-|-2gr

This series is to be continued, by putting N'^'' for the numerator, and
D^""^ for the denominator, of the r^^ fraction; and then computing their
values, by means of the following formulas ;

[8644] J){r) _ £)(r-l) _|_ 5 (r-1)^ />(r-2) .

Then the values of the following fraction,

= 0-f -

[8645] J, 25^ l-L.'—

1+^ ^ • ''"'

which occurs in [8639], will be included between the two fractions

[8646] ■— and _^^^ , Hence we find that f^dx-c is nearly equal to* y'^;

wouiTbe and it follows, from [8619], that if the sun's atmosphere were taken away, it

bruik" would appear twelve times as luminous. This result depends however on

[8647] the experiment of Bouguer, which ought to be repeated several times with

mospheie much carc, upon several points of the sun's disk.

were re- ■*• •*■

* (3986) We have, in [8611, 8620], ?= 7 = =0,7, nearly. Substituting

this in [8641], we get s^^^, s^^^, s^^\ &c. ; hence we can form the series of fractions
[8639/]. The fifth of these fractions is 0,50, and the sixth 0,48 nearly. Their mean
gives nearly F=0,49. Multiplying it by 5 =0,7, and by c - ^ = 0,24 [8612], we
obtain f . ^.c"-'' = 0^08, or yV nearly, as in [8646].

X. iv. ^ 14.] ON THE MEASURE OF HEIGHTS BY A BAROMETER. 565

CHAPTER IV.

ON THE MEASURE OP HEIGHTS BY A BAROMETER.

Symbols.

[8648]
[8648']

14. The measure of heights by a barometer depends, like the theory of
refraction, upon the law by which the density of the strata of the atmosphere
decreases. We shall iise the following symbols ;

a = the distance from the centre of the earth to the lower station of
the observer; ■■'>:^"i<i'u

a-\-r = the distance from the centre of the earth to the upper station of

the observer ;

(^) = the force of gravity at the lower station ; [8648"]

g = the force of gravity at the place of the particle of air ; [8648'"]

(p) = the pressure of the atmosphere at the lower station ; [8649]

p = the pressure of the atmosphere at the place of the particle of air ; [8649']

P = the density of the air at the distance a-{-r from the earth's centre ; [8649"]

z = the heat of a particle of air at the distance a-{-r from the earth's

centre ; and when r = 0, z becomes equal to q [8657, 8661],

Then we shall have, as in [8469, 8653</],

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