23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 171 



expression for the secular acceleration. It is very remarkable, however, that 

 although he finds these values of 8u and Snt, he does not substitute them 

 in his equations, but puts 8u and Snt = instead of them. It is only 

 by this strange process of suppressing part of the results which he himself 

 has found, that M. Plana arrives at a different value of the secular accele- 

 ration from mine. Indeed, in the first form of this Memoir, as I have 

 already mentioned, M. Plana did actually obtain a value coincident with 

 mine. 



M. Plana is led to make this suppression of his own results by a 

 supposed a-prion proof that a certain integral which is equivalent to 



> i 

 2 r j dv 



} dv 



can contain no such terms as those which would arise from the substitution 

 in it of the true values of Su and 8)it. Now, even if this proof had been 

 ever so convincing, M. Plana was surely bound to shew in what manner 

 the terms thus arising from 8u and 8nt were destroyed, as the different 

 parts of his investigation would othei'wise contradict each other. 



In fact, however, this proof is entirely fallacious, for it rests on the 

 assumption made at the top of p. 43 of the Memoir, that the terms multi- 

 plied by p, p~, &c., in the equation given on the preceding page, may be 

 neglected ; and these are precisely the terms which are equivalent to those 

 which M. Plana suppresses. 



It may be as well to make another remark on this part of the in- 

 vestigation. In p. 42, M. Plana puts 



e" cos gr = 1M cos (pv + q}, 

 e" sin gr = HM sin (j)v + q), 



and he assumes that all the coefficients p will be small quantities. But 

 this will not be the case when e'^cosgr and e' g &mgr are thus expressed 

 in terms of the Moon's longitude. If these functions were similarly expressed 

 in terms of the time, viz., if we were to put 



e" cos gr = 1M cos (pt + q), 

 e' ff sin gr 2M sin (pt + q), 



all the coefficients p would be small. 



222 



