710 Mr, A. Stephenson on the Stability of 



the coefficients of sin ?d and cos nt to zero, we obtain 

 u" + 2qv' + (^ - <f - 3a V> - fa 2 {3u 2 + v 2 + it 3 4- uv 2 ) = 0,(1) 

 i*'-2qu'+ {f^-q 1 - a?fi 2 )v-fi 2 ot%2uv + v* + u q v) = 0, . (2) 



a being small the changes in the amplitude and relative 

 phase o£ y take place so slowly that u" and v" may be 

 neglected, and we find by eliminating t and integrating, 



2 (fi 2 -q 2 - a V) O 2 + rV «V (u 2 + v 2 + 2z<) 2 = a constant, (3) 



Since the ultimate motion is independent <>£ the initial 

 small disturbance we may put the constant zero : then 

 initially when u and v are small 



O 2 - q 2 - 3* VK + O* ~ ?* - "> 2 = 0, 

 which is possible only within the limits of instability. This 

 condition takes the simple form 



u/v = + tan j3 : 



thus y is initially either in that relative phase in which it 

 receives the maximum energy, or in the phase of opposite 

 sign in which it gives up energy. 



2a 

 If u = r sin 6 and v=rcos#, y— — «^r sin (^£ + 0), and (3) 



\/3c 



reduces to . n . , N 



sin 0=±(\— r), 



where \* = 2-(ji*-q*-e?p')/Jp*=$(l + k) ; 



and A, is positive for an increasing amplitude. 

 Again, from (1) x v — (2) x u, 



If r is initially positive r'/r increases until r = X, and then 

 decreases reaching zero when r=X + 2, which is therefore 

 the maximum value of r. If r is initially negative the first 

 extreme value is A- — 2. If k is nearly — 1, A, is small and 

 the maximum amplitude of y is twice that of the steady 

 motion : for & = 0, the positive and negative extremes are 

 V2 + 2 and +/2 — 2, and if k is nearly 1, they approximate to 

 4 and 0. Thus in the case of small motions the resultant 

 amplitude may vary numerically from zero to anything up 

 to five times its steady value. Small friction, however, while 

 not greatly changing the character of the motion may narrow 

 these limits considerably. 



Owing to the wide variation in amplitude following a small 



