23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 173 



Now M. Plana, in his Theory of the Moon, supposes the value of the 

 above integral to be 1264""lf*, and in his Memoir in vol. xviii. of the 

 Turin Memoirs he gives it the value 1297"'7f. 



If, then, we reduce the coefficients of the secular equation given by 



these authors, so as to make them correspond with the value 1270" 2" of 



the above integral, which is that employed in my Memoir of 1853, they 

 will become 



Coefficient according to M. Plana's theory 10'60, 



,, M. Plana's memoir (1856) 1T24, 

 ,, M. Hansen's theory 1276. 



The difference between M. Hansen's coefficient and either of M. Plana's 

 is much greater than could possibly have arisen if both values had been 

 found on correct principles, and they had differed merely in consequence of 

 the approximations not being carried far enough. 



My value of the same coefficient, which was communicated to the 

 French Institute in January, 1859, is 5"70. And M. Delaunay, while 

 perfectly agreeing with me in the terms which I have calculated, has added 

 a great number of others depending on the eccentricity and inclination of 

 the Moon's orbit, and thus increases the coefficient to 6"'ll. 



As M. Hansen's method of obtaining his coefficient has not yet ap- 

 peared, it is, of course, impossible for me to point out the reason of the 

 difference between it and my own, as I have done in reference to the 

 results of MM. Plana and de Pontecoulant. I have very little doubt, how- 

 ever, that it arises from M. Hansen having tacitly assumed, like M. Plana, 

 that one of his constants introduced by integration is an absolutely constant 

 quantity. 



M. Hansen has suggested that the difference between his result and 

 that obtained by M. Delaunay and myself may arise from want of con- 

 vergency in the series proceeding according to powers of m, by means of 

 which we determine the coefficient denoted above by K. 



If we confine our attention to the terms of K which are independent of 

 the eccentricity and inclination of the Moon's orbit, and which are admitted 

 by all to constitute by far the largest part of that quantity, we find that 

 the terms involving the successive powers of m taken into account by me 



