168 Mr. A. Stephenson on the Frequency Ranges of 



The case c = \n admits of similar treatment. When c has 

 this value we find that either 



=0, (4) 



— l^-J +a 1 cii + a 2 * 2 + * 3 



ai + ojj 1 ~^(J L ) +** ai + * 4 



«2 + a 3 



CCi + U± 1 



or 



*i 



cc Y — a 2 1 



a 2 — « 3 



«i — a 2 



2 



a 2 — «3 



«! — « 4 





0, (5) 



where the term in the rth row and 5th column is 



except in the case r = s, for which it has the form 



l-(2 S -iy(n/2 f jLy±ot 2s _ 1 . 



Hence the ranges of n associated with 2/x, . . . 2\i\r are 

 obtained. 



Thus when c = equation (1) splits up into (2) and (3); 

 and when c = ^n, into (4) and (5). 



4. When the non-generating force is simply periodic 



x -{- jj?{1 + 2u cos nt) 2 = 0. 



If a. is so small that its square may be neglected, it is 

 evident from (4) and (5) that the range of cumulative effect 

 about the double frequency, 2/x, is 2/i{l + ^a). To a second 

 approximation the limits are given by 



i-(£)' ± . . 



-K0 



= 0, 



whence 



2^1±i«-T 1 6<) 



(6) 



