30 



LECTURES ON THE LUNAR THEORY. [LECT. 



Now these equations are defective, for they have been formed by 

 omitting certain terms from the complete equations as given in Lecture II. 

 Hence, calling /, 8 the values of I, 6, which we have proved in Lecture IV 

 to be solutions of the above equations, if we substitute I,, a in the com- 

 plete equations of Lecture II, residuals are left, say X and Y respectively. 

 And if I, 6 be solutions of the complete equations, and if we write 



where SI, 8/9 are small quantities whose squares and products may be 

 neglected in the first instance, we obtain the following equations for deter- 

 mining 81, SB, the corrections to approximate solutions 1 , already found: 



--. i 

 dt 2 at at (It (It a 



til,, (. . ,, s , 



} -i _i_ 2 -r 2 - + 2 - + 3n cos 2wo0 0. 



df dt dt dt dt 



Now let us write 



= 1 + H , >1 = V* = <> + >, 



" 





where c = ( I + n')~ + - n'\ 



Zi 



The quantity v consists wholly of periodic terms of the form cos 2i 

 multiplied by small coefficients ; w contains, besides periodic terms, a small 

 constant term, which however might be removed if we were to choose c 

 as the constant part of p/r* in place of according to the definition above. 



Let S'l, S'0 be quantities defined by the equations 



then S'l, S'0 are approximations to the complete corrections 8/, 80, which if 

 substituted in the equations that give those corrections will leave residuals, 

 say X' and Y', where 



