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which certainly is not obtained by that from which it is a departure. It would be out of keeping with the rest of the work to have introduced into this part of it any algebraic investigations; else it would have been easy to show that the mode of procedure here followed leads direct, and by steps (for the subject) of the most elementary character, to the general formulæ for these perturbations, delivered by Laplace in the Mécanique Céleste.*

The reader will find one class of the lunar and planetary inequalities handled in a very different manner from that in which their explanation is usually presented. It comprehends those which are characterized as incident on the epoch, the principal among them being the annual and secular equations of the moon, and that very delicate and obscure part of the perturbational theory (so little satisfactory in the manner in which it emerges from the analytical treatment of the subject,) the constant or permanent effect of the disturbing force in altering the disturbed orbit. I will venture to hope that what is here stated will tend to remove some rather generally diffused misapprehensions as to the true bearings of Newton's explanation of the annual equation.f

If proof were wanted of the inexhaustible fertility of astronomical science in points of novelty and interest, it would suffice to adduce the addition to the list of members of our system of no less than eight new planets and satellites during the preparation of these sheets for the press. Among them is one whose discovery must ever be regarded as one of the noblest triumphs of theory. In the account here given of this discovery, I trust to have expressed myself with complete impartiality; and in the exposition of the perturbative action on Uranus, by which the existence and situation of the disturbing planet became revealed to us, I have endeavoured, in pursuance

* Livre üi. chap. viii. art. 67. | Principia, lib. i. prop. 66, cor. 6.

of the general plan of this work, rather to exhibit a rational view of the dynamical action, than to convey the slightest idea of the conduct of those masterpieces of analytical skill which the researches of Messrs. Leverrier and Adams exhibit.

To the latter of these eminent geometers, as well as to my excellent and esteemed friend the Astronomer Royal, I have to return my best thanks for communications which would have effectually relieved some doubts I at one period entertained had I not succeeded in the interim in getting clear of them as to the compatibility of my views on the subject of the annual equation already alluded to, with the tenor of Newton's account of it. To my valued friend, Professor De Morgan, I am indebted for some most ingenious suggestions on the subject of the mistakes committed in the early working of the Julian reformation of the calendar, of which I should have availed myself, had it not appeared preferable, on mature consideration, to present the subject in its simplest form, avoiding altogether entering into minutiæ of chronological discussion.


Collingwood, April 12, 1849.





Of the figure of the Earth. Its exact dimensions. Its form that of equili-

brium modified by centrifugal force. Variation of gravity on its surface.

Statical and dynamical measures of gravity. The pendulum. Gravity




motions. Estimation of the disturbing force. Its geometrical representa-

tion. Numerical estimation in particular cases. Resolution into rect-

angular components. Radial, transversal, and orthogonal disturbing

forces. Normal and tangential. Their characteristic effects. Effects of

the orthogonal force. Motion of the nodes. Conditions of their advance

and recess. Cases of an exterior planet disturbed by an interior. The

In every case the node of the disturbed orbit recedes on the

plane of the disturbing on an average. Combined effect of many such

disturbances. Motion of the Moon's nodes. Change of inclination.

Conditions of its increase and diminution. Average effect in a whole revo-

lution. Compensation in a complete revolution of the nodes. Lagrange's

reverse case.

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