28 Proceedings of the Royal Irish Academy. 



At time t the value of v is given by the two-dimensioned form of (28), yiz. : — 

 2v sinh Ih 

 (l^ + m') B 



sinh Ih sin [Ix + (m - l^t)y) - sinh li}i-y) sin b: - sinh hj sin {/>: + [m - l[5t)b} 



sinh lb sin { Ix - (in + l^t)y] - sinh l{h-y) sin Ix - sinh ly sin { fe - [m+l^t)}) ] ' 



and that of ?/, obtained from dujdx + d.vjdy = 0, is given by 



2lu. sinh Tb 

 {]? + r,i^)B 



-[rn-l^t] sinh/& sin{/r/;+(??i-/j3;^)?/} + / eoshZ(&-?/) cos/a'-^ cosh Z^/ cos {/.r+(?n-/)30^} 

 " ' l^^iin-l^tj " 



{m-^I(5t) sinh lb sin ( b:-[m^-l^t)y] 4 /cosh Z(5 -y) cos /•:>:-/ cosh ly cos { lx-(m^l^t]b } 



z^ + (??? + iisty * 



(38) 



Aet. 6. Reference to End-conditioim; Exo.nvple of Prescribed Conditions. 



This, then, is the solution with the given initial conditions if the dis- 

 turbance is to remain periodic in cc and of the assigned wave-length. It 

 appears desirable, however, to allude to cases in which other and more 

 definite conditions may be assigned at the ends of the stream. Suppose, for 

 instance, with the same initial disturbance, it is made a condition that u 

 should vanish at two fixed planes x = 0, o: = a, perpendicular to the 

 direction of flow, in which case, of "course, we must have sin la = in 

 order that this condition should be satisfied initially. We now add to the 

 values u, v given by (38) others Ui, Vi, which (i) satisfy equations (29), 

 (ii) vanish everywhere initially in the region considered, (iii) make Vi vanish 

 at the planes y = 0, y = b, and (iv) make Wi = - u at the planes 

 X = 0, X = a. These may be obtained as follows : — Denoting the value of w 

 as found from (38) at the planes x = 0, x = a, by ^'.q fy, t), Ua {y, t), 

 respectively, expand these functions by Fourier's theorem in series of the 

 forms 



u^ [y> t) = ^ ^r cos rnylb, 



Ua [y, t) = '2 Br COS rTTyjb \ 

 wherein the values of r are positive integers and A,., B,-, are functions of t 



