23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 165 



Thus the same system of simultaneous simple equations that gives the 

 values of the indeterminate coefficients, determines likewise the value of 



fiffl 



j- , which is what we want to find. 



If the Moon's longitude v be taken as the independent variable, we 

 must proceed according to the same principles, but there is one additional 

 circumstance to be attended to. 



In the former case, since e' is supposed to vary uniformly with the 



de' . d*e f 



time, -j- is considered constant, or -j-^ = 0. In the latter case the terms 



which are introduced by the consideration of the variability of e' will 



de' , de' 



involve ,- instead of - 7 as before; and since the Moon's motion in loncfi- 

 dv at 



tude is not uniform, the value of -7- cannot be considered constant, or -= 



dv dv* 



de' 

 cannot be neglected. To take this into account we must substitute for -y- 



its value -; ,- , in which -j- is a known function of v, and then the 



at dv dv 



remainder of the process will be exactly similar to that before described. 



Let us now consider the method followed in M. Plana's theory, and 

 also by M. de Pontecoulant. 



dd 



In this method the terms above described involving -y- are ignored, 



and consequently the differential equations as developed by these astronomers 



d'Yii 

 furnish no materials whatever for determining the value of -j- . Hence 



they are forced to supply the lack of data by means of an assumption, 

 which is that one of the so-called constants introduced by integration is 

 absolutely constant. 



The value of any one of the constants so employed can be expressed 

 in terms of n, e' and known quantities. If then this so-called constant were 



ClfJ 



really so, we should be able by differentiating this relation to obtain -j- 

 in terms of -- . But if on the other hand this supposed constant be 



