170 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 



and the h~ employed in the Memoir is the absolutely constant quantity added 

 to complete the integral, so that if for the sake of distinction h* be written 

 for the h? of the Memoir, we shall have 



h-=v + h-|-^-y. 



7 .. L !785 



or Ir = hi 11 mV 2 



[ t>4 



The following relation exists between the h of M. Plana and the h 

 of M. de Ponte"coulant : 



h 75 m< e/2 



h 64 



Now this relation at once shews that if e' be variable, h and h cannot 

 both be constant ; and since no a-priori reason can be given why one of 

 these quantities should be constant rather than the other, we are not 

 justified in assuming that either of them is so. 



This argument, however, does not appear convincing to M. de Ponte"coulant. 



In the two methods which, as I mentioned before, I employed previously 



fi/n 



to the publication of my Memoir of 1853, the value of , was deduced from 



those of T 7- and y , respectively. In the method which I now employ, 



y- is determined by direct substitution in the differential equations, without 



introducing either the quantity h or h, that is, without taking into con- 

 sideration the mean areal velocity at all. 



In M. Plana's Memoir, contained in the eighteenth volume of the Turin 

 Memoirs, he no longer maintains the constancy of his quantity h, but he 

 determines its variation incorrectly, only taking into account part of the 

 terms which produce this variation. M. Plana here recognises the reality 



de' 



of the supplementary terms involving -7, which I have proved to exist 



Cvt> 



in the expressions for the Moon's coordinates ; and he finds values for Su 

 and Snt in pp. 14 and 20 of the Memoir, which coincide with mine, except 

 in the terms with the argument c'mv, in which a mistake occurs in his 

 coefficients, which, however, does not affect the coefficient of m 4 in the 



