Orr — Stability or Instability of Motions of a Perfect Liquid. 21 



in this interval. Further, in order that the method may apply to an arbitrary 

 initial disturbance, it is necessary that there should be a series of such 

 quantities Vr, and, associated with each, a function v,. of such a character that 

 an arbitrary function of y can be expanded in a series of the form 



which converges in the given interval. The quantities Vr are required to 

 exist for all real values of k 



" Lord Eayleigh has proved that it is impossible to satisfy the differential 

 equation and the boundary conditions with a complex value of V, if d'^ Ujd'if 

 is one-signed between the boundaries ; and he concluded that, under this 

 condition, the steady motion expressed by U must be stable. It appears, 

 however, that this conclusion required additional justification, inasmuch as 

 there is no evidence to show that every disturbance will be propagated by 

 waves in the manner supposed. Lord Eayleigh has further remarked that it 

 is impossible to satisfy the differential equation and the boundary conditions 

 with any value of V for which U - K and fP Ujdy'^ have the same sign 

 everywhere between the boundaries." 



Professor Love then proceeds to examine a certain example in which 



d^ Uldy"^ is one-signed between the boundaries, and proves that in its case,* 



" though there may be a finite number of values of V for which the 



differential equation 



.^, ^..fd'v . \ dW 



^""-""^df-'V^'W 



has a solution v, which vanishes when y = hi and when y = hi, there cannot 

 be an indefinite series of such values. It follows that, though there may be 

 particular types of disturbance which can be propagated by wave-motion in 

 the manner supposed, this cannot be true for a general disturbance." 



Further on Professor Love refers to the case in which U is a linear 

 function ol y: he writesf : — 



" The differential equation becomes 



5-.'. = 0. (15) 



and a solution vanishing when y = hi is 



V = A sinh k (y - hi), 



but we cannot make it vanish also when y = /iz.. In this case it has been 

 suggested that a possible wave-motion might be found by taking V equal to 

 the value of IT at one line, y = a say, between hi and h^. Then the 



*L. c, p. 207. tL. c, p. 212. 



