26 



Proceedings of the Royal Irish Academy 



instead of X- = P as in the two-dimensioned disturbances considered by 

 Lord Eayleigh. The initial value of v given by (24) is the real part of 

 Be'^^ sin my cos nz ; and when this complex expression is expanded, as far as 

 it involves x and y, by the aid of (22) in the form 



the corresponding value at any time t is obtained by multiplying each 



element of this integral by e-^'^^'^K Thus, on rejecting the imaginary parts, we 



obtain a value for v given by the equation 



cv 

 X sinh Xb.v/B cos nz = sinh X(h -y) (X^ + m^) sinh Xrj sin mi] cos / {x - (5rit) di] 



J 



+ sinh \y (y -f m^) sinh X (5 - rj) sin mr\ cosI{x - (5rjt) drf. (27) 

 Jv 



On performing the integrations this result is seen to be equivalent to 



2v sinh Xh 



(X^ + m") B cos Qiz 



sinh Xh sin { Ix + (m -l^t)y]- sinh Xih-y) sin Ix - sinh Xy sin { Ix + (m -ll5t)b\ 



" X^ + (m - /j30' ^ 



sinX& sm{lx - (m + Ifit) y] - sinh X(b- y) sin Ix - sinh Xy sin { Ix - (w + IjSt) h} 



X- + {m + l(5ty 



(28) 



in which the second member on the right is obtained from the first by 

 changing the sign of 7n, and prefixing a negative sign. 



Art. 5. Preceding Remit oUained more directly ; corresponding value of u in 



two-dimensioned case. 



The above result may, however, be obtained more directly from the 

 fundamental hydrodynamic equations. These take the forms 

 du ^ du ^ 1 dp ^ 



dt ^'^ dx ^ p dx 



1 d2} 

 p df 



dv „ dv 

 dt ^^ dx 



dw -, dw 



dt 



dx 



1 dp 

 p dz' 



y, 



(29) 



du dv dw 

 dx dy dz 



-^ = 



in which, as usual, p denotes pressure, and p density ; and we require 



