198 



For the earth, however, the constant e is quite insensible ; but for the 

 moon, Laplace says that it is variable. However, it will not affect the 

 recent inquiry in either case, and therefore may be dismissed for the 

 present. 



Formation of the Constant Quantities in the Differential Equations. 



If, now, we multiply together the two values of w x and w 2 , given 

 above, we shall have 



6)^2 = {Ax cos {n - n x t + e) + B x sin (n - % t + e)} 



{A 2 cos (n - n h t + s) + B % sin {n - n x t + e) } 

 = \A X A* + BA) + 

 periodic terms ; retaining the constant part 



B-A IB-Af . - \ 



0 1 2 2 C 



or, putting for A u &c, their values, and dividing numerator and deno- 

 minator by 



C-AC-B 



n - n 



1" *>2 



AB 



_ 1 B-A JST.(M y +M x ) 

 ~2 ABC B. 



where 



C-A C-B 



I) = -— n 2 - n - n? 



AB 



This part, therefore, of the differential equation for w 3 contains a 

 constant term ; but before we can say that the entire equation does so, 

 it is necessary to develop the term on the. right side of the equation. 

 Now, if this is expanded, it will be easily seen to consist entirely of 

 sines and cosines, of which the general form may be said to be 



/ sin (pn - qn t t + 



where / is some function of i, and p and q are whole numbers. ' It 

 would appear, therefore, at first sight, to contain no constant term ; 

 but in reality it will be seen that it does. For, it is easily seen that 

 every term in and w 2 , such as those found above, will introduce a 

 periodic term into the value of i, as also of <p and 0, and the multipli- 

 cation together of periodic terms may produce a constant. To see what 

 terms in the development spoken of will be necessary, we must find the 

 variations of i, <}> and 0, then, 



i 



