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



CM Ti 



It is obvious that these several values of. 7- contradict each other, 



nut 



and the reason is that the quantities h, h, and a are really variable, and 

 that therefore ^ y? , y-y- , and -y- have been wrongly neglected. In order to 



f> T) 



find the true value of y- we must therefore determine the values of these 



nat 



last-mentioned differential coefficients, and substitute them in the several 



dn 

 expressions tor y- given above. 



de' 



Now the supplementary terms involving -y- which I have shewn to 



ct/t 



exist in the expressions for the Moon's coordinates, will introduce into the 

 integral 



2 \r~-j-dv, 

 J dv 



besides periodic terms, a non-periodic one of the form 



-j' dt, or He'\ 



consequently, since in the equation 



I* 



. j Ctv\ . , \MJ.V j 



r -rr = h + 2 \i*-T- dv, 

 \dt I j dv 



M. Plana considers h" to denote the whole of the non-periodic part of 



i* ( -yr ) , h 2 must consist of an absolutely constant part together with the 



\cttj 



variable quantity He'* just mentioned 



^ 

 and .'. jfcr must be equal to H 



^ 



Similarly -y-- may be found by determining the non-periodic term which 

 is in the same way introduced into the integral 



* 



dv 



in the equation 



.dv , dR 



