708 Mr. A. Stephenson on the Stability of 



solving this completely, but it is evident that there is always 

 a particular solution of the form 



« = i{A f .Bin(2r + l)gt + B l .cos {2r + l)qt}. . (ii.) 

 o 



giving the steady forced motion following the requisite 

 initial conditions. Oar problem is equivalent to the question, 

 in what circumstances is this motion unstable ? 



To determine these we put x = Q+y, where Q represents 

 the right-hand side of (ii.) ; (i.) then reduces to 



y' + 2«y + /x 2 (l-3 C Q 2 -3oQ^- C /)y=0, . (iii.) 



and if the disturbance from the steady state is small, the 

 initial motion is given by 



y' + 2/^ + /* 2 (l-3cQ% = 0. . . . (iv.) 



When Q 2 is developed this becomes 



y" + 2tcy'+fjL 1 2 {l — frjcos {2qt + e x ) — & 2 cos (4gtf + e 2 )...}y = 



where ft 1 2 = yu, 2 (l— the mean value of 3c*Q 2 ). 



Thus the spring of y is subject to an imposed periodic 

 variation, and therefore, k being sufficiently small, an oscil- 

 lation of frequency r times that of Q is continually forced if 

 fjb^g has any value within a certain range in the vicinity 

 of r, where r is any integer *. If fju^q does not lie within 

 any of these ranges y = e~ Kt z, where z is made up of simple 

 periodic elements, so that the small disturbance is damped 

 out, and the steady motion is stable. 



In the cases of instability the magnifying effect is most 

 marked when q is nearly equal to /a 1? the variation in spring- 

 having then the maximum influence in neutralizing and 

 overcoming the damping due to friction : we shall examine 

 this in detail. 



2a 



2. If a is sufficiently small we may put Q= — -r=cos<^, 



where a is small, and (iv.) becomes v^c 



y / + 2*/ + / u 1 2 (l-2a 2 cos2^3/ = 0, . . (v.) 

 where /x 1 2 = yLt 2 (l — 2a 2 ). 



If \jjii — q \<^oi 2 q the solution is 

 y = *-«*{ A*^ 1 -*^ sin (qt + 0) +~Be- 1 ^ l -^ t sin (qt-/3)} 



where ixi — q — \ka?q and tan /3= +\/ • 



* "On a class of forced oscillations," Quarterly Journal of Pure and 

 Applied Mathematics, no. 148 (1906). 



