Precession.

Fig. 7.

35

Various modes of

The equator EAC, by taking the position BAD, recedes from the ecliptic in the colure of the solstices application. CL, and CD is the change of obliquity of the nutation. For let CL be the solstitial colure of B AD, and c/ the solstitial colure of EAC. Then we have sin. B: sin. E sin. LD: sin. lc; and therefore the difference of the arches LD and le will be the measure of the difference of the angles B and E. But when BE is indefinitely small, CD may be taken for the difference of LD and lc, they being ultimately in the ratio of equality. Therefore CD measures the change of the obliquity of the ecliptic, or the nutation of the axis with respect to the ecliptic.

36

37

38

the momentary nutation will be r 31 kpxy. In this
3 tkp
value is a constant quality, and the momentary
2 T
nutation is proportional to a y, or to the product of the
sine and cosine of the sun's longitude, or to the sine of
twice the sun's longitude; for ry is equal to half the
sine of twice %.

If therefore we multiply this fraction by the sun's
momentary angular motion, which we may suppose, with
abundant accuracy, proportional to %, we obtain the
fluxion of the nutation, the fluent of which will ex-
press the whole nutation while the sun describes the
arch of the ecliptic, beginning at the vernal equi-
nox. Therefore in place of y put /1-x2, and in place
of % put
and we have the fluxion of the nu

x

tation for the moment when the sun's longitude is %, and the fluent will be the whole nutation. The fluxion 3 tkpxx, of which the resulting from this process is

tkp

2 T

fluent is 3px. This is the whole change produ

4 T

The real deviation of the axis is the same with the change in the position of the equator, P p being the measure of the angle EA B. But this not being always made in a plane perpendicular to the ecliptic, the change of obliquity generally differs from the change in the position of the axis. Thus when the sun is in the solstice, the momentary change of the position of the equator is the greatest possible; but being made at right angles to the plane in which the obliquity of the ecliptic is computed, it makes no change whatever in the obliquity, along the arch ecliptic, reckoned from the vernal ced on the obliquity of the ecliptic while the sun moves but the greatest possible change in the precession. equinox. When this arch is 90°, is 1, and thereIn order to find CD the change of obliquity, observe that in the triangle CAD, R sin. AC, or R: cos. fore 3 tkp is the nutation produced while the sun moves AE sin. A sin. CD,=A: CD (because A and CD 4T are exceedingly small). Therefore the change of ob- from the equinox to the solstice. liquity (which is the thing commonly meant by nutation) CD=Ax cos. A E, r km n, cos. A E'=r 3P

:

:

3
2 T

The momentary change of the axis and plane of the equator (which is the measure of the changing force) is

3tk
2 Tmn.

41

The momentary change of the obliquity of the eclip- The real 31 k Pxx tic is

2 T

and momentary

changes

greatest at

the solsticrs, and at

Precession, tity of the precession after a given time, that is, the arch BE for a finite time.

42

Quantity of precession in a given time.

43 Mode of using the formulæ.

44

We have ER: CD=sin. EA: sin. CA (or cos.
EA)=tan. EA: 1, and EB : ER=1: sin. B. There
fore EB CD-tan. EA : sin. B. But tan. EA=
sin. long.
cos. long.

cos. Ex tan. ES, cos. EX

qx

=

=p, and CD EB:

Therefore EB : CD =

qx

sin. obliq. eclip.

If we now substitute for CD its

3tkp
21

tan. long. O
value found in N° 40. viz.
3 kqxx

t

27X

2T

-xx, we obtain EB

the fluxion of the precession of the

Log P
Log 5".292

We must now recollect the assumptions on which Assumpthis computation proceeds. The earth is supposed to tious on be homogeneous, and the ratio of its equatorial diame- which the ter to its polar axis is supposed to be that of 231 to 230. If the earth be more or less protuberant at the ceeds. equator, the precession will be greater or less in the ratio of this protuberance. The measures which have been taken of the degrecs of the meridian are very inconsistent among themselves; and although a comparison of them all indicates a smaller protuberancy, nearly Tinstead of, their differences are too great to leave much confidence in this method. But if this figure be thought more probable, the precession will be reduced to about 17" annually. But even though the figure of the earth were accurately determined, we have no authority to say that it is homogeneous. If it be denser towards the centre, the momentum of the protuberant matter will not be so great as if it were equally dense with the inferior parts, and the precession will be diminished on this account. Did we know the proportion of the matter in the moon to that in the sun, we could easily determine the proportion of the whole observed annual precession of 504" which is produced by the sun's action. But we have no unexceptionable data for determining this; and we are rather obliged to infer it from the effect which she produces in disturbing

√√1—**), the first is incomparably the regularity of the precession, as will be considered

greater than the second, which never exceeds 1", and is
always compensated in the succeeding quadrant. The
3tkq
precession occasioned by the sun will be
%, and
4T
from this expression we see that the precession increases
uniformly, or at least increases at the same rate with
3tkq
the sun's longitude ≈, because the quantity is con-
4T

are the sine and cosine of 23° 28', viz. 0.39822 and plane of the lunar orbit, in the same manner as the so-
lar precession is reckoned on the ecliptic. We must

47

Fig. 8.

mx t

T

Cos. E
Cos. 231

09

TX cos. r

Precession. the angle E contained between the equator and the lu- the angle N, which may be considered as constant dur- Precession. ing the month, and the longitude N, which is also nearnar orbit, the precession will be = ly constant, by observing that sin. E: sin. ❤N=sin. N: mat sin.N sin.VN •cos.E sin. E. Therefore RS=" But we must exterminate the angle E, because it changes by the change of the position of N. Now, in the triangle VEN we have cos. E=cos. Y N · sin. N·sin. Y— cos. N cos., y ca-db. And because the angle E is necessarily obtuse, the perpendicular will fall without the triangle, the cosine of E will be negative, and we shall have cos. E-bd-acy. Therefore the nutation mx t cx (bd—acy) for one month will be = X the " T b

48 Lunar pre

a month

reduced to

tic.

and it must be reckoned on the lunar orbit.
Now let B (fig. 8.) be the immoveable plane of
the ecliptic, EDF the equator in its first situation,
before it has been deranged by the action of the moon,
AGRDBH the equator in its new position, after the
momentary action of the moon. Let EGNFH be the
moon's orbit, of which N is the ascending node, and
the angle N=5° 8' 46".

Let Ny the long. of the node be
Sine NV

Cosine Ny

In order to reduce the lunar precession to the ecliptic, cession in we must recollect that the equator will have the same inclination at the end of every half revolution of the the eclip- sun or of the moon, that is, when they pass through the equator, because the sum of all the momentary changes of its position begins again each revolution. Therefore if we neglect the motion of the node during one month, which is only 1 degrees, and can produce but an insensible change, it is plain that the moon produces, in one half revolution, that is, while she moves from H to G, the greatest difference that she can in the position of the equator. The point D, therefore, half-way from G to H, is that in which the moveable equator cuts the primitive equator, and DE and DF are each 90°. But S being the solstitial point, S is also 90°. Therefore DSE. Therefore, in the triangle DGE, we have sin. ED: sin. G=sin. EG : sin. D, EG: D. Therefore D=EGX sin. G, EGX sin. E. nearly. Again, in the triangle DA we have sin. A sin. D (or cos. E) sin. D : sin. ♥A,=D:~A. Therefore D cos. E EG sin. E cos. E,

=

mt sin. E cos. E cos. E

T

49

Nutation in the

sin. 23

This is the lunar precession produced in the course of ac3). Now this is equal to mn_(db × versed

one month, estimated on the ecliptic, not constant like
the solar precession, but varying with the inclination or
the angle E or F, which varies both by a change in the
angle N, and also by a change in the position of N on
the ecliptic.

We must find in like manner the nutation SR pro-
duced in the same time, reckoned on the colure of the
same time, solstices RL. We have R sin. DS=D : RS, and
RS-D sin. DS, D⋅ sin. E. But D=EG sin. E.
mt cos. E
Therefore RS EG⚫ sin. E sin. ~ E, ='
T cos. Y
X sin. Ex sin. E. In this expression we must substitute

2

e

sine, %-acX versed sine 22): For the versed sine
of x is equal to (1- —cos. ≈); and the square of the sing
of an arch is the versed sine of twice that arch.

This, then, is the whole nutation while the moon's
ascending node moves from the vernal equinox to the
longitude vN=≈. It is the expression of a certain
number of seconds, because, one of its factors, is the
solar precession in seconds; and all the other factors are
numbers, or fractions of the radius 1; even e is expres-
sed in terms of the radius I.

The fluxion of the precession, or the monthly preces-
sion,

53

54