196 



and density with respect to the planes which contain its principal axes, 

 it is manifest that all quantities of the form 2 (m x x p z x q ), &c, when 

 either p or q are odd numbers, will vanish. Thus, suppose q to be an 

 odd number, then any particle m having % x for one of its co-ordinates 

 will always be accompanied by another having - z x for one of its co- 

 ordinates; hence the sum of these will vanish. Now, these are the 

 sort of quantities, both in the example which I have chosen, namely, 

 terms depending upon sin 0 - 0 and cos 0 - 0, and in all other cases 

 whatever where constant terms can be produced. Where this is the 

 case, therefore, the quantities in the equations for w l9 &c, on the right 

 side will vanish ; but this will not be the case for such bodies as the 

 earth, moon, &c, whose form, though nearly spherical, &c, differs 

 from it on account of the irregularities of surface, &c. 



Integration of the Differential Equations. 



For this purpose the following equations for determining Q, &c, 

 which are given in all dynamical treatises of motion about a fixed point, 

 will be useful, viz. : — 



6^0 cos* , . 



— = cj 3 — : — la x sin <p + <w 2 cos <p) 

 at sin i 



# 1 . 



— — = : (<y, Sin ® + &> 2 COS <2>) 



at sin i 

 di 



— = u x cos <p - a 2 sin <p, 

 ut 



in which \J/ is the longitude (measured backwards) of the moveable axis 

 of referred to a fixed line. 



The first of them will give us an approximate value for 0 : to use 

 it we must first find w 3 . ]S"ow, neglecting the disturbing force, as also 

 products of io Y vo 2 , which are supposed very small, and are, moreover, 

 multiplied by B - A, the third of equations {A) becomes 



du % 



— = 0, or u z = n 

 at 



Neglecting, therefore, small quantities in the first of the equations of 

 this number, it becomes 



— - n, or <p - nt, 

 at 



if the time is supposed to commence when the R. A of the example x is 

 0 ; also for 0 it will suffice to put w, + e. Thus the equations for » x 

 and a. 2 become, on putting n for a z , nt for <p, &c, 



