to a higher order. It is then found that the next term in the coefficient is m3, which is about of the first term; and as there are several other important terms, it is only by carrying the approximation to a higher order (the 5th at least) that the value of this coefficient can be obtained with sufficient accuracy from theory. In fact, m* would give a coefficient of 28' 32" only; whereas the accurate value is found to be 39' 30".

The same remark applies also to the coefficients of all the other terms.

74. As far as terms of the second order, the coefficient of the variation is independent of e the eccentricity, and k the inclination of the orbit. It would therefore be the same in an orbit originally circular, whose plane coincided with the plane of the ecliptic: it is thus that Newton has considered it. Princip. Prop. 66, Cor. 3, 4, and 5.

Annual Equation.

75. To explain the physical meaning of the term

– 3me' sin(mpt + B − 5)

in the expression for the moon's longitude.

0=pt — 3me' sin(mpt + B − 5),

=pt — 3me' sin (longitude of sun – longitude of sun's perigee), =pt - 3me' sin (sun's anomaly).

Hence, while the sun moves from perigee to apogee, the true place of the moon will be behind the mean; and from apogee to perigee, before it. The period being an anomalistic year, the effect is called Annual Equation.

Differentiating we get

do

dt

=p{1-3m2e' cos (sun's anomaly)}.

Hence, so far as this inequality is concerned, the moon's angular velocity is least when the sun is in perigee, that is at present about the 1st of January, and greatest when the sun is in apogee, or about the 1st of July.

The annual equation is, to this order, independent of the eccentricity and inclination of the moon's orbit, and therefore, like the variation, would be the same in an orbit originally circular. Vide Newton, Principia, Prop. 66, Cor. 6.

Reduction.

76. Before considering the effect of the term

k2 4

sin 2 (gpt — y),

which, as we shall see Art. (82), is very nearly equal to the difference between the longitude in the orbit and the longitude in the ecliptic, it will be convenient to examine the expression for the latitude of the moon, and to see how the motion of the node is connected with the value of g.

LATITUDE OF THE MOON.

77. The expression found for the tangent of the latitude,* Art (49), is

s = k sin (g0 − y) + 3mk sin {(2 − 2m − g) 0 − 2B+y}. If we reject all small terms, we have

8 = 0,

or the orbit of the moon coinciding with the ecliptic, which is a first rough approximation to its true position.

*This expression for the tangent of the latitude is more convenient than that which gives it in terms of the mean longitude, Art. (53) on account of the less number of terms involved. See Pontécoulant, vol. iv., p. 630.

Take NN' (g-1) in a retrograde direction, and join MN' by an arc of great circle;

then

or

sin N'm=tan Mm cot MN'm,

sin [0 - {y — (g−1) 0}]=s cot MN'm;

which, compared with the value of 8 given above, shews that MN'm = tan1k is constant, and therefore the term k sin (g✪ — y) indicates that the moon moves in an orbit inclined at an angle tank to the ecliptic, and whose node regredes along

de

the ecliptic with the velocity (g-1), or with a mean velocity (g-1) p.

79. Hence the period of a revolution of the nodes

but, from Art. (49), the value of g=1+&m2;

This will, for the same reason as in the case of the apse, Art. (68), be modified when we carry the approximation

to a higher degree; this value of g is, however, much more accurate than the corresponding value of c, for the third term of g is small; the value to the third order being (see Appendix, Art. 95)

This is not far from the accurate value as given by observation, and when the approximation to the value of g is carried to a higher order, the agreement is nearly perfect. The true value is 6793.39 days, that is about 18 yrs., 7 mo.

Evection in Latitude.

80. To explain the variation of the inclination and the irregularity in the motion of the node expressed by the term

+ 3mk sin {(2 − 2m − g) 0 − 2ß + y}.

This term, as a correction on the preceding, is analogous to the evection as a correction on the elliptic inequality. Taking the two terms together,

s = k sin (g0 − y) + ğmk sin {(2 − 2m − g) 0 − 2B+y}.
)= longitude of moon = 0,

Let

therefore

if

s = k sin()—8) + 3mk sin { ) — 8 — 2 (0 - 8)}. Now these two terms may be combined into one

K2 = k2 {1+§m cos2 (☺−8)}2+k2 {}m sin2(0— S)}2,

or approximately,

8=3m sin 2(-8),

K=k(1+m cos2 (0-8)};

but the equation

s = K sin()-8 - 8)

represents motion in an orbit inclined at an angle tan 1K to the ecliptic, and the longitude of whose node is 8 +8.

This term has therefore the following effects:

1st. The inclination of the moon's orbit is variable, its tangent increases by mk when the nodes are in syzygies, and decreases by the same quantity when they are in quadrature; the general expression for the increase being mk cos2(0-8).

2nd. The longitude of the node, calculated on supposition of a uniform regression, is increased by d= 3m sin2 (○ — X), so that the node is before its mean place while moving from syzygy to quadrature and behind it from quadrature to syzygy. Principia, book III., props. 33 and 35.

The cycle of these changes will be completed in the period of half a revolution of the sun with respect to the node, that is, in 173-21 days, not quite half-a-year.

81. The tangent of the latitude has here been obtained; if we wish to have the latitude itself it will be given by the formula

which, to the degree of approximation adopted, will clearly be the same as s.