To find M-Suppose a small arbitrary rotation given to the planet through an angle dy about the axis of y, that is, about a line through L the centre of gravity of the planet, in the plane of maximum areas perpendicular to LM its line of nodes: let the effect of the rotation be to change the position of this plane from MI to mi: draw Mn perpendicular to mi.

Now (Art. 10 (iii)) ¥1+g=f(h, k, 8),

and 0, 4, do not involve g; therefore

motion. Any other supposition might be adopted with regard to the motion of this plane, but this is the most convenient, and will be retained throughout this Article.

To find N:-Suppose a small arbitrary rotation given to the planet about the axis of z, that is, about a normal through the centre of gravity of the planet to the plane of maximum areas: then we may suppose, alone to vary, and since the tendency of N is to diminish,, we shall have

as in the preceding Article; then the third equation gives

16. The equation for calculating y may also be simply obtained as follows: if we refer the motion to the fixed plane of reference, and suppose V expressed as a function of 0, 4, , since k cos y is the area conserved on this plane, we shall have

Now (Art. 10 (ii)), † — a =ƒ (01, ¥1, y), and 0, & do not involve a; therefore

the latter differential coefficient supposing V expressed as a function either of 1, 1, Y1, α, y, or simply of t and the

The method of this Article is identical with the second method given in Art. 34 of the Planetary Theory for the dedi

termination of; we reserve the comparison of the results to a subsequent Article (Art. 22).

dl

17. The following proposition will be useful in finding dt'

being connected together by the equations of Art. 10, (iii).

where s =

Co,. Then from Arts. 3 and 6, we have

(1),

(2),

From equation (2), by differentiating under the integral

= √(AB) [k (u-v) ds, by equations (1).

(uv)+

From equation (3), in like manner, we have

To simplify this expression, we find from equations (1),

18. The following relations between partial differential coefficients of the function V, as well as the results of the

preceding Article, will be required in finding

dl

dt

In Art. 10 we have shewn that, in order to express V as a function of t and the elements, we may first express it as a function of 0, 1, 1, α, y, by case (ii), and then eliminate ,,,,, by means of the equations in (iii). The first three of these equations are