Orr — Stability or Instability of Motions of a Perfect Liquid. 39 



Fixing attention on the first and second terms alone on the right, and 

 writing them in the form 



ry rh 



sinh \(b -y) IT cos I (x - ^r]t) dt] + sinh \y Vcosl (x - j3r)^) dri, (50) 



Jo J y 



on integration by parts, since the terms at the limits cancel, this becomes 



mr 



y dU [^ dV 



sinh \{h -y) —^ sin / {x - (3t)t) dti + sinh \y -r- sin l{x- jSijO dx] 

 dri 



y dri 



(51) 

 On integration again by parts, we obtain terms at the limits varying as t--, 

 which do not cancel, and also integrals which, when t is sufficiently great, may 

 be proved to be negligible in comparison with those terms. 



The third and fourth terms of v may be treated similarly ; and the result 

 stated as to the ultimate form of the value of v thus follows. 



And, in the two-dimensioned case, the corresponding value of to at time t 



is evidently given by 



ry - cb 



2u = coshX (b-y) U &m I (x- (irft) dri - cosh. Xy \ V sin I (x - ^rit) dn 

 Jo J y 



+ two other terms. (52) 



On integration by parts in the same manner, we obtain terms at the 



limits which do not cancel, and vary ultimately as t"^, and integrals which, 



when t is large enough, may be neglected in comparison with those terms. 



Akt. 13. Case of Severed Layers of Constant, hut Liferent, Vorticities. 



I proceed to allude briefly to the more general case, in which the 

 stream is composed of a number of layers, each having constant, but 

 different, vorticities, and there being no slipping at the surfaces of transition. 

 Equation (31) holds for each layer; and its first integral throughout may be 



written 



Sj''v = F{x-Ut,y,z), (53) 



U being in any layer of the form /3y + c, with different values of j3, c in 

 each layer. If we take a two-dimensioned disturbance, in which initially 



Vq = 2 cos Ix sin my/(P + m^) (54) 



we have, at time t, 



y2^ = - sin {l(x- Ut) + my] + sin [l{x- Ut) - my]. (55) 



For brevity, consider only the first term ; this, of course, corresponds to a 



wave which might occur alone. This leads to, in any layer, 



s\Ti.\l(x- Ut)-\-my\ , ,_„ ■ 



