846 Mr. A. Stephenson on the 



3. In the case of an alternating impulsive field 



x + 2*n 2 1 cos (2r + L)nt . a?=0. 

 o 



* 2 4 



Equation (i.) gives 2a=lfZ n q \2 = ^2? an( ^ (^0 nas 



no positive root. Hence there is instability if the impulsive 

 spring is not less than four times the reciprocal of its period. 

 This result may be obtained directly. If a point with one 

 degree of freedom is subject to changes of velocity /3 x dis- 

 placement and — /3 x displacement at intervals t, then if the 

 initial velocity is properly chosen relatively to the displace- 

 ment, the displacements at intervals 2t increase or decrease 

 in geometric progression provided (3t>2, the ratio being 



_i( /S V-2±/3rv / iSV^4). 



The displacements at the impulses are 



#, =f ka.v, —k' 2 ^, ±k z ax, k A x, ... , 

 where 



h=±(fi T:t 4 /£V , -4) and a= \/(j8t-2)/08t+2). 



The general solution is of form (1) where the period is 4r, 

 and 



-c/)(2t-O = 0(O = {1-(1-#i)^/tK 



+ {l — (l + ak 1 )tJT}e-P t between and t, 



^(2r-t)=^(t) = \l-(l-alk l )tlr}e^ 



- \ 1 — (1 + ah^t/rle-P* between and t, 



&! being the larger value of k, and p=- log c ^= - cosh -1 /3r/2. 



In the limiting case /3t = 2, the particular solutions are 



A 1 l (O and Bi^^O—^iCOK 

 where fa and ^ are even and odd functions of t defined by 

 — ^(27- — t) = fa(t) = l — t/r between and r 3 

 ^i(2t- 0=^i(0 =^( 3 ~ 2 ^/ T ) between and r. 



These give motions in which the displacements at the 

 impulses are b, 0, — b, 0,b . . . and 0, — c, — 4c, c, 8c, . . . , results 

 which are easily verified. 



The case in which the impulses are of the same sign may 

 be treated similarly, Since in the above the steady motion at 

 the limit of stability is unaffected by the omission of the 



