95, 96] ROTATION OF THE EARTH. 299 



But this is equal and opposite to the action of the sun upon the 

 earth, the moment of which about the earth's center of mass is 

 accordingly 



161) M=-(eX-xZ), 



Differentiating the expression 159), since x appears both explicitly, 



and implicitly in r, and -^- = ? 



J dx r 



dv^dv^ 



dx dr r 



162) | F = |I^ + 



oy or r 



fa + 



cV dV z_ 

 r 



and inserting in 161), 



L = - 



163) 



We may now insert these in Euler's equations, so that, if x, y, z, 

 the coordinates of the sun, are given as functions of the time, the 

 earth's motion may be found. Considering the earth to be symme- 

 trical about its axis of figure, we put A = _B, so that N = ,0, and 

 the third equation gives r = const., as in the case of the top. It is 

 however more convenient for our purpose to use, instead of Euler's 

 equations a set of equations proposed by Puiseux, Resal, and Slesser, 

 in which we take for axes, as suggested in 84, the axis of symmetry, 

 and two axes perpendicular to it, that is, lying in the equator, and 

 moving in the earth. We have, since we are dealing with principal axes 



164) H x = Ap, H, = Aq, H z =Cr, 



which are to be inserted in equations 29), 84, where we are to' 

 put the velocities with which the moving axes turn about themselves, 

 which we will call ^ , g , r , so that our equations are 



165) 



dH, 



dt 

 dH 



dH 



