29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 237 



which are those of the ordinary lunar theory, in which the motion of the 

 ecliptic is not taken into account, so that x, y, z may be supposed to be 

 known functions of t. 



Hence the equations for determining the small increments Sx, 8i/, Sz 

 of the coordinates, which are due to the motion of the ecliptic, are the 

 following : 



d- Sx . m' ~ SLLX , , 5. , 3m V 



ty = -^r- 



'I 



- , dz 



We may remark that no terms involving arbitrary constants need be 

 added to the values of Sx, By, 82, since these may be supposed to be already 

 included in the values of x, y, z. 



Hence we may choose for Sx, Sy, 82 any particular values which satisfy 

 these differential equations, and we may consider these values to contain w 

 as a factor throughout. 



If y denote the sine of the mean inclination of the Moon's orbit, the 

 value of z, and therefore that of ,- , will contain y as a factor throughout. 



Hence the form of the first two of these differential equations shews that 

 the values of Sx, 8y, found under the above conditions, will contain yw as 

 a factor throughout, and therefore that the term 



which occurs in the third differential equation, will contain the factor y"(a 

 throughout. 



If, therefore, we neglect the square of y, the equation for Sz takes 

 the simple form 



d-8z ' /x 



