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



really variable, we must take its variation into account, in order to obtain 



the true value of -j- in terms of ^-. 

 at at 



In M. Plana's theory, in which v is taken as the independent variable, 

 the constant so employed is h 2 , which is added to complete the integral 



f . dR 1 . . , 

 2 \r~ -v dv, in the equation 

 J dv 



- 



dv 



in w 

 of v. 



C d f? 

 hich 2 I r -j dv is supposed to consist of a series of cosines of multiples 



The quantity r 2 -3- is equal to twice the area described in a unit of 



Gvtt 



time, or to twice the areal velocity, so that h 2 is the non-periodic part 

 of the square of twice the ax-eal velocity, the periodic part being supposed 

 developed in cosines of multiples of v. 



In M. de Pontecoulant's theory, the constant h is introduced to complete 

 the integral l-r-dt in the equation 



dv (dR , 



in which -y- dt is supposed to consist of a series of cosines of multiples 

 of t. 



M. de Pontecoulant's h is not identical with M. Plana's h, but there 

 is a simple relation between these quantities. 



M. de Ponte'coulant, however, does not employ the constant h in finding 

 value of the secular acceleration, but anoth 

 introduced to complete the integral in the equation 



the value of the secular acceleration, but another constant - , which is 



a 



2 df r a J dr 



all the periodic terms of which are supposed to consist of cosines of multiples 

 of t. 



