204 



No account, however, has as yet been taken of the first terms which 

 occur in the development of L, namely, 



3 fi B-A 



2 r* 



a 2 sin 2<p - 29 + 2afi sin 2<p + /3 2 sin 20 + 20 



Now, it is evident that these will contain constant terms, which, if 

 they do not identically destroy each other, or are destroyed by terms 

 which may arise in other parts of the differential equations, will give 

 terms indicating a gradual and permanent change of motion. Let us 

 see how constant terms might arise in the above expression. 



In the first place it has been already seen that the expression for i 

 contains, amongst others, terms of the form 



H sin 2n - n\ IT cos n 1 , &c, 



we shall thus have 



cos i = cos (« x + iZ"sin 2n - n l + iTcos = cos {i l + P) 



suppose 



= cos *i (1 -\P % + 7) + sin J P - — - 7 



= cos i t { 1 - iZ"sin 2n - w> JTcos n 1 } + P sin <„ &c. 



= cos «i { 1 - i ^TXsin 2n - 2n l + ± JTJTsin 2n}+P sin i x . 



In like manner if sin (2$ - 29) be developed, it will contain the 

 term sin 2n - 2n\ terms multiplied by P, and constant terms. The 

 first of which, when multiplied by HK sin 2n - 2n l , will produce a 

 constant term, and the second when multiplied by P sin «"„ and the 

 third when multiplied by cos i v 



Again, the first approximate values already found for i x <p and 9, 

 will, by substitution in the differential equations, produce terms having 

 the arguments 



2n - 2n^ 2n, &c, in 1 <p and ty. 



And these will arise in two ways : first, it is evident that such a term 

 as H cos n 1 occurring in the expression for % would, when introduced 

 into a or cos i, where it occurs in the terms in the differential equations 

 already used for determining terms in &c, having the argument 

 <p - 9 or n - n l , would introduce terms into '»„ &c, having the argu- 

 ment n - 2n lt and these, when multiplied by cos <p, where it occurs in 

 the equation for i, viz. u x cos <p, would produce terms having 2n - 2n l 

 for their argument in the expression for i. These terms, however, will 

 be multiplied by higher powers of sin i than those which arise in the 

 manner about to be examined, and therefore for a first approximation, 

 at least, may be neglected, especially in cases where i is small ; and 



