300 



VII. DYNAMICS OF ROTATING BODIES. 



If we choose as X-axis the line of nodes, or intersection of the 

 equator with the ecliptic, or plane of the sun's orbit about the earth, 

 we have, in 65), cp = 0, so that Euler's geometric equations become 

 simply, 



1 a<z\ dft d"' m ' 7 " N 



Ibb) Po = ~ / j7' <l* = - 



We have also P=PQ, # = #o> while r is n t equal to r Q . Inserting 

 in the third equation 165), we have C = 0, r = const. = &, where 



i& is the angular velocity of the earth's daily rotation. 



We shall content ourselves with an approximate solution of the 

 equations, which may be obtained by neglecting the squares and 



products of small quantities. Observations show that -^ and -r- are 



/ dib \ 



small, l-j-j- = 50",37 per year), so that we may neglect r g , 

 Thus our equations 165) become 



167) 



If the sun, or other disturbing body, did not move with respect to 

 the axes of X, Y", Z, then Z, M would be constant, and the equations 

 would be satisfied by constant values of j?, #, 



168) 



M 



In order to ascertain whether these approximations are sufficient 

 when L and M vary, let us differentiate equations 167), substituting 

 in either the value of the first derivative of p or q from the other, 

 obtaining 



169) 



dt* 



A 

 CO, 

 A 



^. T ^ 



4- M) = 



dL 



~dt"' 



dM 



dt 



Fig. 107. 



We have now to find the values of L, M 

 in terms of the motion of the sun. 



If I be the longitude of the sun, that 

 is the angle its radius vector OS makes 

 with the X-axis, we have, passing a plane 

 through the sun perpendicular to the X- axis 

 (Pig. 107), 



x = r cos I, y = r sin I cos #, 

 = r sin I sin -fr, 



