ORDER OF DISTURBING FORCE.

23

24. The differential equations in Art. (20), when these values of the forces are substituted in them, would contain a new variable ', but we shall find means to establish a connexion between t, 0, and e' which will enable us to eliminate '.

They will, however, be still incapable of solution except by successive approximation; but before proceeding to this, it will be important to consider the order of the disturbing effect of the sun's action, compared with the direct action of the earth. Now, if we examine the values of P, T, and S, it will be found that the most important of the terms containing m', which are clearly the disturbing forces since they depend upon the sun, are involved in the form while those independent of the

m'r

13 1

Now the orbits being nearly circular, and m the ratio of the mean motions, Art. (22), we have

or the disturbing force of the sun is of the second order.

CHAPTER IV.

INTEGRATION OF THE DIFFERENTIAL EQUATIONS.

SECTION I.

General process described.

25. The differential equations which we have obtained are, as already stated, incapable of solution in their general forms; and even when P, T, &c. have been replaced by their values, the integration cannot be effected, and we must proceed by successive approximation.

Firstly, neglect the disturbing force of the sun which is of the second order, and also the moon's latitude, which, as will be seen by referring to the expressions for the forces (23), will either enter to the second power or else in combination with the disturbing force.

When this is done the equations can be integrated, and values of u and s obtained in terms of correct to the same order of approximation as the differential equations themselves, that is, to the first order; and this value of u will enable us to get the connexion between 0 and t to the same order.

[Let us, however, bear in mind that the equations thus integrated are not the differential equations of the moon's motions, but only approximate forms of them; and it is, therefore, possible that the results obtained may not be even approximate forms of the true solutions.

Whether they are so or not, can only appear by comparing them with what we already know of the motion from observation, and this previous

GENERAL PROCESS OF INTEGRATION.

25

knowledge, in the event of their not being approximations, will probably suggest such modifications of them as will render them so.*]

The integration of the equations (a), (B), (v) can be performed when the second members are circular functions of 0; and as the first approximation will give us the values of u and s in that form, when these values are admissible and carried into the expressions for the forces, they also will be expressed as functions of 0, and we can proceed to a higher approximation.

The new approximate values of P, T, S are then made use of to reduce the second members of the differential equations to functions of 0, retaining those terms of the expressions which are of the second order.

The equations are again integrable, and this being done, the values of u, s, t will be obtained correctly to the second order. These values introduced in the same manner in the second members, and terms of the next higher order retained, will lead to a third approximation, and so on, to any order; except that if we wish to carry it on beyond the third, the approximate values of the forces, given in Art. (23), would no longer be sufficiently exact.

26. There is, however, a peculiarity in these equations, when solved by this process, which we must notice. We have said that to obtain the values to any order, all terms up to that order must be retained in the second members: but it may happen that a term of an order beyond that to which we are working would, if retained, be so altered by the integration as to come within the proposed order.

Such terms must therefore not be rejected, and we shall proceed to examine by what means they may be recognised.

dou d02

the same form as that of + u = (0), even though (0) should be a very

approximate value of ƒ (u, 0), but there is a presumption in favour of such a supposition.

27. Suppose then that after an approximation to a certain order, the substitutions for the next step have brought the equation in u to the form

where the coefficient G is one order beyond that which we intend to retain. The solution of this equation will be of the form

u = + G' cos(p0+ H) + ·······

......

G' being a constant to be determined by putting this value of u in the differential equation,

G
1 - p* '

29

from which we learn that if p differs very little from 1, G' will be one order lower than G, and will come within our proposed approximation, and consequently the term G cos(p◊ + H) must be retained in the differential equation.

The equation of the moon's latitude being of the same form as that of the radius vector, the same remarks apply to it.

28. Again, in finding the connexion between the longitude and the time (one of the principal objects of the Theory), wemust use equation (a), Art. (20),

Now, having developed the second member and substituted for u, &c. their values in terms of 0, let it become

Therefore, when q is of the first order, will be one order lower

than Q, and the term will have risen in importance by the integration.

T

10.

increased in value; for they increase once in forming de,

and once again, as above, in finding t.

T

h2u3

Since such terms occur in the development of and also

39

de of on account of their previously being found in u, we must dt 9 examine how they appear in the differential equation that gives u, that we may recognise and retain them at the outset. Now, by referring to the last article, we see that when

G and G' will be of the same order; and in

P is very small

1 T

u2

29

u3

the order

of the term will still be the same; so that all the terms which it will be necessary to retain are known at the outset.

29. We have, therefore, the following rule:

In approximating to any given order, we must, in the differential equations for u and s, retain periodical terms ONE ORDER beyond the proposed one, when the coefficient of 0 in their argument is nearly equal to 1 or 0; and terms in which the coefficient of the argument is nearly equal to 0, must be retained TWO ORDERS higher than the proposed approximation when they occur in

T

3.

h2u3 •

If we wished to obtain u only and not t, there would be no necessity for retaining those terms of a more advanced order in which the coefficient of 0 nearly equals 0.

SECTION II.

To solve the Equations to the first order.

30. We shall in this first step neglect the terms which depend on the disturbing force, i.e. those terms which contain m', for we have seen, Art. (24), that such terms will be of the second order.