182 Professor Baker, On the stability of periodic 



But for w = 3, there is stability unless H lie between 



Pi2A2-PA3, lk^^\^ + P\\ {P = lk^^~U^k^ + l'^), 



whicli limits do not include zero unless \ == 0. For greater values 

 of n the range of values for H within which stability fails is that 

 between two values of the form 



-2— T ^i''^' + ^^^ -2^-~i '^I'A' + -^A'. 

 n^ — 1 n^ — \ ^ 



(See Royal Sac. Phil. Trans., A, ccxvi, 1916, 184.) 

 In the following we consider a system of equations 



dt ~dy/ dt dx/ ^ ^,^,-..,ri), 



wherein the function F is expressible as a convergent power series 

 in a small parameter /x, 



F=F,+ i.F^+..., 



whose coefficients are analytical functions in 



X-^, ..., X^, y-^, ... y^, 



having no singularities in the range of values here considered, and 

 are periodic, of period 27r, in each of ^i, ..., y^ separately. Thus F 

 does not contain t explicitly. It is further assumed that Fq is a 

 function of x-^, ..., x^ only. This is an important Umitation, sug- 

 gested by the use of Delaunay's variables in the problem of three 

 bodies. We shall for the most part limit ourselves to the case 

 when w = 2, so that there will be four differential equations. 

 If x-^ ... x^ be initial values for x-^ ... x^, the quantities 



'*" ~ die,' 



for Xg = x^, may be called the conventional mean motions. By 

 Poincare's theory there is a periodic motion provided certain re- 

 strictions for the initial circumstances are introduced, when the 

 ratios n-^ : n^ : ... are commensurable — that is when there are 

 {n — 1) identities nja2 — n^a^ = 0, n-^a^ — n^a^ = 0, ... with integer 

 coefficients a^, a^, .... By appropriate linear transformation of the 

 variables aj^, x^, ..., ^j, y^, ..., these conditions are reducible to the 

 simpler forms % = 0, ^2 = 0, ..., these being n — I in number. 

 More precisely, noticing for the sake of comparison that, for ju, = 0, 

 the differential equations are satisfied by 



Xi = Xj^, x^ = x^, y^ - n^ + tui, ^2 = ^2^ + ^2. 

 where w-^, w^ are arbitrary, it can be shown that, for small values 

 of fx, there is a periodic solution, reducing f or i = to 



x^ = x^ + A^, x^ = x^o + A^, yi=mi + B^, ^3 = t^g + B^, 



