186 Professor Baker, On the stability of 'periodic 



As we have computed A, B, C, D only to the first power of /x, 

 we retain only to first power of /n in this determinant. It is con- 

 venient to multiply by the determinant 



which for n variables x and n variables y would be equal to p^", and 

 for n = 2 is equal to p^. Then the above determinant becomes 



- p (/xa - p), - pfi^ 



p, {HQa - py), p2 _ ^p§ _|_ ^Hq^ 



and the evaluation to the first power of /x is extremely simple, 

 namely in general it is 



where M is the sum of the diagonal elements of the two matrices 



- pa, - /)§ + Hq^, 



which in general are each of n rows and columns. 



Removing the factor p^'^, which was introduced, and remem- 

 bering that the roots p are in pairs, equal, and opposite in sign 

 (cf. p. 165 of the paper referred to), we infer that the sum of the 

 diagonal elements of a and S is zero, and to our approximation the 

 equation reduces to 



which however, as was remarked above, divides by p^. Thus we 

 see that for n = 2, p'^/p, is developable as a power series in /x, and 

 further that the sign of p^ is that of the negative of the sum of the 

 diagonal elements of the matrix, of two rows and columns, ex- 

 pressed by Hq^. 



Now consider the meaning of j8. We have 



dy^ 

 where we are to put 



^1 = ^i> Vx = Vi, Vx = n^t + tui, y^ = n^t + w^, 

 and, as was said, we have n^ = 0. In general F^ has a form 



^x = Ao + ^\p, cos (^i?/i + ^2^2 + h), 

 wherein Aqq, Aj.^p,^, h are functions of x^^, x^. Let [F-^] be the part 

 of F;j^ which is a function of ^^ only, 



[Fx] = Ao + 2^p,./os (^1^1 + h), 



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