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and putting therein eft in the place of e', we have the first column of the foregoing expression of ę.

The second column, involving sin arg., contains the new periodic terms considered in Prof. Adams' Memoir of 1853, and the coefficients for the arguments g', 27, 27 −g', 2r+g', agree with his values; observing that his terms belong to -de, so that the signs are reversed; those for the remaining arguments 2g', 2r-2g', 2r+2g', are not given by him.

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The equation for dv may be written,

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192, 193

n2

Mr. CAYLEY: on the Secular Acceleration of the Moon's Mean Motion.

and the part m212 (C + S¿Qdt) contains (see Annex 23)

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194, 195

that n2 a3 Sum of the masses of the Earth and Moon; or, taking this to be unity, we have N2 A3 = 1, n2 a3 = 1.

The formula of Prof. Adams' memoir, which it will be necessary to make use of, may be written

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It will be convenient to write M, N, A, E', in place of the m, n, a, e', of the foregoing part of the present Memoir, and to now use m, n, e', in the significations in which they are employed by Prof. Adams; E' (the constant part of the solar excentricity) is his E', and his e' is E'+f't. As to his symbols a, a,, these, I think, ought to have been represented, and I shall here represent them by a, a,.* And I take a such

* Plana, in his Lunar Theory, uses the three letters a, a,, a; his a and n Sum of the masses of the Earth and Moon. There is being such that n2 a3 an obvious inconvenience in writing a, a,, in the place of his a, a,.

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1785 5355 128 128

= and that it is in fact by the

make together 3. 2187 addition of these terms that Plana's coefficient is changed 128 into

3771 64

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VI.

If the investigation were pursued further. a question would arise as to the proper form to be given to the arguments; for in these, nt seems to stand in the place of v, the value whereof is

v = n t + −

(23 m2 - 3771 m2 ) ne' f't,

64

say v = n t + + k ne' ft2, and it might be considered that in the arguments nt+ should be changed into nt++kne'f" t2, or, what is the same thing, that should be changed into +kne' f't, but that g' should remain unaltered (this assumes that there is not in the Sun's longitude any term corresponding to the acceleration). The arguments, instead of being of the simple form kt, would thus be of the form kt + kft. But this would not only increase the difficulty of integration, but would be inconsistent with the general plan of the solution; and it would seem to be the proper course to imagine the cosine or sine of such an argument to be developed

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COS sin

(cos kt + k2 ft2

2.

= k tk, f' t kt in such manner as to bring the secular part of the argument outside the cos or sin; this is, in fact, the form which the solution takes when the arguments are left throughout in their original form, for the terms of the form f't arg. would present themselves in the subsequent approximations. But I shall not at present further examine the question.

COS

sin

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(3153

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39

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