188 Professor Baker, On the stability of periodic 



however involves that the characteristic exponents are in pairs 

 of equal values of opposite sign. There is therefore also another 

 solution of vanishing characteristic exponent. 



Now suppose that in the argument of p. 162 of the paper referred 

 to the matrix Q" (u) has not Hnear invariant factors. It will be 

 sufficient to take one case, and to suppose that 



?'C.ia) . ?'f.,(. 



e ' , icoe 



Noticing then, multiplying the matrices, that 

 x-\ . , . X', ., . 



. , F, it'Y' 



• , ., T 



X'X-\ . , 



. , Y'Y-\ i{t' -~t)Y'Y-^ 



Y' y-i 



of which a particular case is for X'= X, Y'= Y, t'= t, we have 



q;+^ (u) = al"' (u) ^; {u) ^ aj {u) ^; {u), 



because u is periodic, and hence, by the form assumed for O!^^ {u). 



, Y-\ ~itY-^ 



O" [u) . k 



-ictico+f) 



, e , — }[cc) -{- t)e 



is equal to 



O^ {u) k 



-icit 



showing that the matrix 

 qI {n) = QlI {u) k 



— ic«f. ■, —ic^t 



e ' , — %te 



- iczt 



- iCit 



— icd ., —ic-it 



. , e ' , —lie 



— ie-t 



e 



k-^ 



has the period oj, and is therefore such that 



Q';{u) = Q'^{u)=^h 



