68 Lord Rayleigh on the Question of 



vibrations can be expressed in a periodic form the steady 

 motion from which they are deviations cannot be unstable. 

 A very simple case suffices to prove that such a principle 

 could not be admitted. The equation to the motion of the bob 

 of a pendulum situated near the highest point of its orbit is 



g-»A,.=X,. (21) 



where X is an impressed force. If X=cos pt, the corre- 

 sponding part of x is 



*=-^£L ; (22) 



p l + mr 



but this gives no indication of the inherent instability of the 

 situation expressed by the free "vibrations," 



x=Ae mt + Be~ mt (23) 



As a preliminary to a more complete investigation, it may 

 be worth while to indicate the solution of the problem for the 

 two-dimensional motion of viscous liquid between two parallel 

 planes, in the relatively very simple case where there is no 

 foundation of steady motion. The equation, given in Lord 

 Kelvin's paper, for the motion of type e i{nt+kz) is 



The boundary conditions, say at <#=+a, are that it, (v) } 

 and w shall then vanish, or by (7) that 



u = 0, du/dcc=0. 



The following would then be the proof from the differential 

 equation that for all the admissible values of w, p is zero and 

 q is positive. 



Writing as before, w=a + i/3, and separating the real and 

 imaginary parts, we find 



-KS-^+KS-^-'lS-'*' 3 )- ' • < 25 » 



Multiply (25), (26) by a, /3 respectively, add and integrate 

 with respect to x over the range of the motion. The coeffi- 

 cient of q is 



