motions in general dynamics 187 



which for y^ = tUj, y^ = n^ + tOg, does not contain t, and let F^' 

 be the remaining part of F^. 

 We have 



^Vr^ys ^Vr^Vs ^Vr^Vs ' 



and > , ^- - vanishes unless r = s = 1, while Q ^ -,s— is an entirely 



periodic function not having the factor t outside trigonometrical 

 terms. Thus in QB the only term having the factor t arises from 



Q ^il^] ; this is 



d'^\F ] 

 - Q^A^jn,, Pi^ cos (p^w^ +h)^- tllA,,,,, p^^ cos (pjW^ + h)^t -^^ 



so that j8 = 



, 



'1 



and S (^0^),, = g^^„, g^^2 ' 



which gives, for the characteristic exponents which do not vanish. 



And it will be recalled that 



In order to obtain the corresponding expression when there 

 are three variables x, and three variables y, it is necessary to carry 

 the approximations as far as ju.^. Poincare's corresponding work is 

 in Meth. Nouv., i, pp. 201-226. 



Note. It has been remarked that two of the characteristic 

 exponents are zero, and that the condition that, in case of equal 



roots, the matrix Q.^^ (u) should have linear invariant factors, is 

 not necessary for the application of the rule above. In fact as t 

 does not occur explicitly in F, the existence of a periodic solution 

 X,. = (j)^ {t), y^ = ifj^ {t) of the original equations, involves the ex- 

 istence also of a solution x^ = (f)^ {t + h), y^ = iftr {t + h), in which 

 h is arbitrary. This involves the existence of a solution 



. _ d(f>^ {t + h) _ difjr (t + h) 



^' dh ' "^^ " dh 



of the equations of variation, where after differentiation h is to be 

 put zero. This is however a purely periodic solution with zero 

 characteristic exponent. The nature of the equations of variation 



