23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 163 



his letter in the Monthly Notices, he appears to have expected that the 

 results of the two methods would differ from each other. One of the 

 values which M. de Pontecoulant thus obtains agrees with that given in 

 M. Plana's theory, as of course it must do, being found by means of the 

 same principles. But he seems to be quite unaware that this value has 

 been abandoned by M. Plana himself in his last paper above referred to, 

 which is contained in the eighteenth volume of the Turin Memoirs. 



M. Hansen's value of the secular acceleration is not given in an 

 analytical form, like those of MM. Plana and de Pontecoulant, and therefore 

 we can only compare the final numerical results. This comparison, which 

 I shall presently give, shews that M. Hansen's value of the acceleration 

 considerably exceeds either of those found by M. Plana. 



Here then we find nothing to inspire confidence ; certainly nothing like 

 the cumulative testimony which there is in support of M. Delaunay's result 

 and mine. 



I may now be permitted to make some remarks on another point. In 

 the introduction to my Memoir of 1853, I gave some general reasoning to 

 shew that a change in the eccentricity of the Earth's orbit had a tendency 

 to produce a change in the mean areal velocity of the Moon, and that 

 M. Plana was therefore wrong in assuming this velocity to be constant, as 

 in his theory he does. Now this seems to have led some persons to 

 imagine that my analysis in the following part of the memoir depended 

 in some way or other on the validity of the general reasoning which had 

 gone before, and therefore that my conclusions could not be regarded as 

 established with mathematical strictness. But this is quite a mistaken view 

 of the case. I make no assumption respecting the variability of the mean 

 areal velocity. I prove mathematically that this velocity does vary by 

 finding the amount of its variation, and the general reasoning given in the 

 introduction is simply the translation, so to speak, of my analysis into 

 ordinary language, in order to make the nature of my correction to M. 

 Plana's theory more generally intelligible. It may be remarked too that 

 even if I had started with the assumption that the mean areal velocity 

 was variable, no error could have been caused thereby, for if this velocity 

 had been really constant I should have found its variation equal to zero. 

 In mathematics the terms " constant " and " variable " are not looked upon 

 as opposed to each other, but a constant is regarded as a particular case 

 of a variable quantity. 



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