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VII. On the Theory of the 31oon. By John William Lubbock, Esq. V.P. and 



Treas. R.S. 



Received November 30, — Read December 12, 1833. 



JM. POISSON having lately published a very important memoir on the Theory 

 of the Moon, I am induced again to lay before the Society some remarks on this 

 subject. 



In this memoir M. Poisson expresses the three coordinates of the moon, namely, 

 her true longitude, her distance, and her true latitude, in terms of the time. The 

 reasons which he adduces for so doing are the same which led me also to deviate 

 from the course which had always been pursued by mathematicians up to the time I 

 commenced the investigation, and which consisted in employing the equations in 

 which the true longitude is the independent variable. 



Instead, however, of integrating the equations of motion by the method of inde- 

 terminate coefficients, as I have proposed, M. Poisson recommends the adoption of 

 the method of the variation of the elliptic constants. Having reflected much upon 

 this question before I entered upon the investigation, I will venture now to state the 

 reasons which determined me not to employ the latter method. 



It seems, in the first place, desirable to introduce uniformity in the methods em- 

 ployed in the theories of the perturbations of the moon, and of the planets, as far 

 as this can be done without the sacrifice of any facility in the solution of the pro- 

 blem. It is not probable, however, that the tables of the planets will be deduced 

 from the variations of their elements. In fact, as I have shown in a former paper, 

 although the results obtained by either method are identical (as is also obvious a 

 priori), it is only by numerous reductions that those deduced from the one method 

 are convertible into those deduced from the other. Moreover, in order to obtain, 

 through the variations of the elliptic elements, the inequalities of any given order in 

 the coordinates, the development of the disturbing function must be carried one step 

 further ; so that, for example, in the theory of the moon, to obtain all the inequalities 

 depending upon the fourth power of the moon's eccentricity, it would be necessary 

 to obtain the terms depending upon the fifth power of the same eccentricity in the 

 development of the disturbing function. 



In the theory of the moon it is necessary to develop many terms in the disturbing 

 function depending on the square of the disturbing mass, and even some depending 

 on the cube ; or, in other words, it is, as is well known, insufficient to substitute, in 



r2 



