450. But zr,s=r, tan o sin (v, - 0), by article 435, or substituting the values of r, and v, in article 447, and rejecting the product e tan, it becomes

and m' on the ecliptic. ecliptic at the epoch;

and 'being the inclinations of the orbits of m These values of z and z' are referred to the but if the orbit of m at the epoch be assumed to be the fixed plane, = 0, tan ' = Y, the mutual inclination of the orbits of m and m', then II being the longitude of the ascending node of the orbit of m' on that of m,

consequently the terms of R depending on z' with regard to y2, ey3, and e'y, become

No. 7. cos {i (n't — nt + c' — e) + 2nt + 2e − 201},

+

2

+ m' . Q. . y'e'. cos {i (n't — nt+c' − e)+3nt + 3e − w' – 201},

4

+ m2

.Qs. y'e. cos {i (n't — nt+e'—e)+3nt+3e — — 211}.

4

451. It appears from this series that the sum of the terms independent of the eccentricities and inclinations of the orbits, is

which is the same as if the orbits were circular and in one plane. The sum of the terms depending on the first powers of the eccentricities has the form

m'
2

E. M cos {i (n't — nt + e' — e) + nt + e + K}.

Those depending on the squares and products of the eccentricities and inclinations may be expressed by

m' Σ N. cos {i (n't — nt + e' − e) + 2nt + 2c + L}

m'

+ Σ N'. —
.cos {i (n't nt + e − e) + L'}.

2

Those depending on the third powers and products of these elements

are

m' & Q. cos {i (n't — nt + e − e) + 3nt + 3e + V}

4

m'

4

&c.

Σ. Q'. cos {i (n't — nt + e' — e) + nt + e + U'},

&c.

It may be observed that the coefficient of the sine or cosine of the angle has always the eccentricity e for factor; the coefficient of the sine or cosine of 2 has for factor; the sine or cosine of 3 has e3, and so on: also the coefficient of the sine or cosine of has tan. for factor; the sine or cosine of 20 has tan2. for factor, &c. &c.

* R

Determination of the Co-fficients of the Series R.

452. In order to complete the developement of R, the coefficients A, and B., and their differences, must be determined. Let

(a2aa' cos ẞ + a2)~A~= 14, +4, cos B

+A,. cos 23 + &c.

The differential of which is

A2saa' sin BA, sin ẞ + 24, sin 23 +34, sin 38+ &c. multiplying both sides of this equation by A, and substituting for A, it becomes

2saa' sin ẞ{4, + A, cos B + 4, cos 2ẞ + &c.}

= (a" — 2aa' cos ß + a3) {A, sin ß + 2A, sin 28 + &c.} If it be observed that

when the multiplication is accomplished, and the sines and cosines of the multiple arcs put for the products of the sines and cosines, the comparison of the coefficients of like cosines gives

in which

(i −1) (a2+a12) A (i-1) — (i+8—2) aa' A(-2);

(i − s) aa'

(119)

may be any whole number positive or negative, with

the exception of 0 and 1. Hence 4, will be known, if A。, 4, can be found.

Let A¬~1 = 1⁄2 B。 + B1 cos ẞ + B2 cos 28 + &c. multiplying this by

and substituting the value of A' in series

4.+A, cos ẞ + 4, cos 2ẞ + &c.

1

=(a3-2aa' cos ẞ + a2) ( B。 + B, cos ẞ + B2 cos 2B + &c.) the comparison of the coefficients of like cosines gives

B¡ ·

A,= (a + a'). B, aa'. B-1)-aa' Bu+1)

B(i+1)·

But as relations must exist among the coefficients B(i−1), Bi, B(i+1),

similar to those existing among A-1), A, A(+1), the equation (119) gives, when s + 1 and i + 1 are put for s and i,

B(+1) == i (a2 + a'2) B; — (i + s). aa' B(-1)

i

(i

If this quantity be put in the preceding value of A, it becomes

2saa' B (i-1) — 8 (a2 + a'2) · B¡ ;

(120)

s

(121)

(122)

whence may be obtained, by the substitution of the preceding value

of B(+1)

▲ (i+1) =3(¿+8) ⋅ aa'(a2+a') B(-1)+s { 2(i −s) aaa" — i(a2 +a12)2}B;

If B be eliminated between this equation and (121), there will B(-1)

1 (i + s) (a2 + a”) A; − 2 (i − s + 1) . aa'. A (+1)

8

(a12 - a2)

or substituting for A(+1) its value given by equation (119),

(i + s − 1). aa' ▲ (i−1)

If to abridge = a, the two last equations, as well as equa

a a'

tion (119), when both the numerators and the denominators of their several members are divided by a2, take the form

(i − 1) (1 + ∞2) A (i−1) − (i + 8 − 2) . a . A (6-2),

(is) a

s

(i − s + 1) a'. A (i+1)

(123)

; (124)

(125)

All the coefficients A,, A., &c., B., B1, &c., will be obtained from equations (123) and (125), when A, A, are known; it only remains, therefore, to determine these two quantities.

e being the number whose hyperbolic logarithm is unity; therefore a2 — 2aa' cosß + a2 = {a' — ac2N=I}. {a' — ac¬3N−1}

consequently,

but

1)

(a' = ac2N=1)~ = _—_{1+ $ ac2N=1+ $(a + 1) œœ2?N=1 + &c.},

2

whence it appears that 1, and i have always the same coefficients; and as disc-12 cos iß, it is easy to see that this series is the same with

=2

A~' = (a'9 -2aa' cos Ba2)~' = ↓ A。 + A, cos B + &c. consequently,

rapidly when s =

1.

2 3

but they converge

then, however, A, and A, become the first and second coefficients of the development of

Let S and S' be the values of these two coefficients in this case,