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VIII. On the Theoi-y of the Moon. By John William Lubbock, Esq. F.P. and 



Treas. R.S. 



Received and Read March 13, 1834. 



fT HEN I commenced the investigations relating to the theory of the moon which 

 I have had the honour to communicate to the Society, I proposed to show how, by a 

 different but more direct method, the numerical results given by M. Damoiseau might 

 be obtained. The approximations were in fact carried much further by M. Damoiseau 

 than had been done before, and the details which accompany M. Damoiseau's work 

 evince at once the immense labour of the undertaking, and inspire confidence in the 

 accuracy of the results offered. But the state of the question is now changed by the 

 appearance of M. Plana's admirable work, entitled " Th6orie du Mouvement de la 

 Lune," in which, although M. Plana employs the same differential equations as those 

 used by M. Damoiseau, and obtains in the same manner finally the expressions for 

 the coordinates of the moon, in terms of the mean longitude by the reversion of 

 series, yet M. Plana's expressions have a very different analytical character and im- 

 portance, from the circumstance that the author develops all the quantities intro- 

 duced by integration, according to powers of the quantity called m, which expresses 

 the ratio of the sun's mean motion to that of the moon. In this form of the expression 

 the coefficients of the different powers of m, of the eccentricity, &c., are determinate, 

 as are, for example, the numerical coefficients in the expression for the sine in terms 

 of the arc, and other similar series. An inestimable advantage results from this pro- 

 cedure, which more than compensates for the great increase of labour it occasions, 

 by diminishing the danger of neglecting any terms of the same order as those taken 

 into account, and by affording the means of verifying many terms long before final 

 and complete results shall have been obtained independently by myself or any other 

 person. By treating the differential equations in which the time is the independent 

 variable, as I have proposed, similar results to those of M. Plana may be obtained 

 directly ; but the calculations which are required in either method are so prodigiously 

 irksome and laborious, that until identical expressions have actually been obtained 

 independently, to the extent of every sensible term, the theory of the moon cannot, I 

 think, be considered complete. It might, indeed, be supposed that already, through 

 the labours of mathematicians, from Clairaut to the present time, the numerical 

 values of the coeflicients of the different inequalities were ascertained with sufficient 

 accuracy for practical purposes, and that any further researches connected with the 

 subject would be more likely to gratify curiosity than to lead to any useful result. 



