To calculate the value of c to the third order.

94. We must here make use of the results which the approximations to the second order have furnished; but as the value of c is determined by that term of the differential equation whose argument is, co-a, we need only consider those terms which by their combinations will lead to it without rising to a higher order than the fourth.

We shall simplify the arguments by omitting 0, α, B, which can easily be supplied by remarking that ce – a and me+B always enter as one symbol, c and m will therefore be sufficient to distinguish them. This only applies to the arguments.

We have, Arts. (48), (23),

u = a {1+e cos (c) + m2 cos (2−2m) + 15 me cos (2 - 2m - c)},

P

h2u2

= a− ¿m2a {1 +3 cos (2 – 2m)} {1 — 3e cos(c).

= a+ &m2ae cos(c) + 13,5 m3ae cosc),

11

32

- §m2 sin (2 – 2m) {1 – 4e cos (c) — 4m2 cos(2 — 2m)

- §m2 sin(2−2m)+3m2e sin(2−2m−c) + 45 m3e sin(c),

· {— §m2 sin (2 — 2m)} {— 15mae sin(2 — 2m — c)}

hu3 de = sin(2

do

+ {3m2e sin (2 — 2m — c)} {— 2m3« sin (2 — 2m)}

= 11⁄2m3ae cos(c),

the other term is of the fifth order,

T

h2u3

d'u

d0 = 3m2 cos (2—2m) — 3m3e cos(2—2m—c) — 45 m3e cos(c),

+u)=2a{1+...—3m* cos(2− 2m) +1m2e cos(2—2m−c)};

45m3ae cos(c).

= a {1+ (3m2e + 1,3,5 m3e — § 5m3e + 45m3e) cos(c)+.....}

= a {1+(3 m2e + 225 m3e) cos (c) +.....}.

Assume u = a {1+e cos(c) +...}

therefore

therefore

;

ae (1 — c2) = (§m3e + 225m3e) a ;

c=1-3m2 - 225m3.

To find the value of g to the third order.

95. This is to be obtained in a very similar manner from

=

+8=&c. We shall, in the argument, write

s=k{sin (g) + ğm sin (2 — 2m-g)},

Whence
Ps-S

72. 3

ds

de

m2k {1+ cos (2 – 2m)} {sin (g) + 3m sin (2 — 2m — g)}

· m3k (1 m) sin (g),

= k cos (g) + 3mk cos (2 - 2m - g);

T ds

therefore

h'u de

- m3k sin(g),

96. Hence, to the third order of approximation,

mean motion of арзе 1 -c 3 m2 +225 m2 8+75m mean motion of node

=

=

=

g-1 3m2 - 331⁄23 8-3m and since m = nearly, we see that the moon's apse progredes nearly twice as fast as the node regredes.

In the case of one of Jupiter's satellites, m is extremely small, for the periodic time round Jupiter is only a few of our days, and the periodic time of Jupiter round the sun is 12 of our years, and therefore m, the ratio of these periods, very small.

is

Hence, the apse of one of Jupiter's satellites progredes along Jupiter's ecliptic, with pretty nearly the same velocity

as the node regredes, assuming these motions to be due to the sun's disturbing force; they are, however, principally due to the oblateness of the planet.

Parallactic Inequality.

97. In carrying on the approximations to a higher order, it is found, as we stated Art. (55), that the expressions for

a' Since

a

E+M a

=

sin(0-0) respectively.

o, nearly, is of the second order, these terms

are of the fourth order, but the coefficient of ◊ being near unity, they will become important in u, and therefore in 0, Art. (27).

We can easily obtain the terms to which they give rise in the values of u and 0,

Substituting in the differential equation for u, we get

98. The corresponding term in the value of 0 will also be of the third order,

T

3

1+. +15m

de is of the fourth order and will not rise in t;

This term, whose argument is the angular distance of the sun and moon, is called the parallactic inequality on account of its use in the determination of the sun's parallax, to which purpose it was first applied by Mayer by comparing the analytical expression of this coefficient with its value