186 



Mr. S. S. Hough. 



[Jan. 16, 



where u, v, w denote the displacements at the point x, y, z; ^ is a 

 function depending on pressure and potential at this point, and n = 

 tt/7>, where it is the rigidity and p the density. 



Putting u, v, w, each proportional to e ikt , and performing the 

 differentiations with regard to the time, the above equations become 



(11 V 2 + A 2 ) u + 2ivi\v 

 (n v 2 + A 2 ) v — 2wiXu 



Ox 

 dy 



>■ 



(2)- 



(wy 2 +\ 2 )w — ^ 



du ^ dv ^ dw 

 dx dy ' dz 



= 



J 



These equations are theoretically sufficient to determine u, v, w t ty, 

 subject to certain boundary conditions. The boundary conditions 

 should also lead to a series of values of A, which correspond to the 

 different periods of free oscillation of the system. 



2. The boundary conditions express the fact that the free surface 

 is not subject to stress. Denoting by cos a, cos /3, cos 7 the direction 

 cosines of the normal to the mean surface, from this condition we 

 obtain the following equations, which hold good at the undisturbed 

 surface : — 



f du , du du du , dv 



cos a -f n < — cos x + — cos /3 + — cos 7 H — — cos « + cos B 

 f [dx dy dz ax dx 



, dw "I 



f dv dv dv 



COS fi + il < -r- COS 2 + 7-COS/3+ -7- cos 7 + 



dy ' dz 



dw \ 



(V-^cos 



du dv 

 — cos a + — cos 3 

 dy dv 



(v'—g£) cos ,3 



f dw , dw „ , dw , tZw efo _ 



y cos 7 + w < — cos a+— cos p + — cos 7+— cos a+— cos p 



(3), 



, dw 



where g denotes the value of gravity at the surface, £" the height of 

 the surface waves, and v' the potential due to the harmonic inequalities. 



