22 Proceedings of the Royal Irish Academy. 



differential equation [(15), above] need not be satisfied when y = a. We should 



then have to take 



V = A sinh k{y - hx), a > y > hi, 



V = B sinh k {h^, - y), hi> y > a. 



To make v and dvjdy continuous a,t y = a, we should require 



A sinh k{a - hi) = B sinh k (Aj - a), 

 A cosh k{a - hi) = - B cosh k [hz - a), 



and these cannot be satisfied when hi is different from hz. Thus there would 

 be in this case no disturbance which could be propagated by waves in the 

 manner supposed. Yet the example afforded by initial disturbances 



to = C(2y - hi - hi) cos kx, 

 V = kC (y - hi) [y - h^) sin kx, 



shows that some varied motion is possible which initially is periodic in o: 

 with given wave-length. Lord Eayleigh's method does not avail for the 

 discovery of this motion, nor for determining whether the original steady 

 motion is stable for this type of disturbance." 



And in the introduction to his paper he expresses the opinion that* 

 " the general conclusion seems to be that wave-motions of Lord Eayleigh's 

 type can only occur in some very special cases, and that his method does not 

 avail for the determination of a criterion of stability when the disturbance is 

 of a general character." 



Art. 3. Remarks on Love's Criticism. 



It appears to me that the remarks which I have quoted embody two 

 misconceptions, and that as a consequence the mathematical investigations in 

 Professor Love's paper are in great measure irrelevant. 



In the first place, we are not entitled a priori to impose the condition 

 that in a perfect fluid dvjdy is continuous across a plane parallel to y. This 

 condition is equivalent to requiring dujdx to be continuous, and therefore 

 either that tt is continuous, or that any discontinuity in it is independent 

 of a; : it involves then either that there is no slipping, or that there is some 

 restriction on its amount ; but we cannot control slipping in a perfect fluid. 

 Whenever the continuity of dvjdy is secured, I apprehend it is by the inte- 

 gration of equation (7) across the plane in question, as Lord Kayleigh has 

 stated, and thus, wherever n + kU ot n {V - U) is zero, discontinuity 

 is permissible. 



Again, we have no right to say that the possible values of V or - njk 



* L. c, p. 199. 



