The Rev. Brice Bronwin on the Theory of Flanetury Dlsturbanve . 33 

 functions of the cosine of Kqt + €„ — tt;, and its multiples ; and assume 



7 = 5 + e5, + ri?2 + • ■ • • 



By substitution, and equalling separately to nothing the terms multiplied by 

 the different powers of e, we have 



-T^ + Uo-B = 0, -j-4 + tto'-Si + £^1 -T-^ = 0, &c. 

 d-r d-r d-r 



The first of these gives B — c cos (jiot + k). This value substituted in the 

 second, it will give Pi ; and so on. But in reducing the periodicals to one ar- 

 gument, we should have a series of sines and another of cosines, unless we make 

 ^ = e,, — TTo. In this case we must have co — 'To = ^i — '^,^ and making n^ = 

 », (1+6), we must develope thus : 



COS (?ZoT + eo - "^(i) = cos (??,T + fo - ■^o) - i«,T sin {ri^r + to - T!„) - &c. 



The use of this is, by suitably determining h, to take away an improper 

 term. In some cases, perhaps, both sines and cosines may be necessary. 

 ]5ut perhaps this will be more conveniently done thus : 



-^=1-2 11 + ^0 cos (Xo-7r„)[, — =^ {1 + e^ cos (X„o- TT,)!. 



Po "0 /',,o ", 



But \,_o = Xo- Therefore we must have wo = tt^, or we should have both 

 sines and cosines. And then we shall have 



— - 1)- =cos (\o - TTo) = cos (\„o - T,) =i~ 1)-; 



or, after a little reduction, 



V _ 1 _ ^ _ Jk_ 5) 



But Po = Pi.a^ts- Substituting this value, we easily find 

 1 _ /i ^«\ f^ , ^'/ ^0 



or, 





VOL. XXII. 



