(7') 



112 Sir William Thomson on Gravitational 



or, in polar coordinates, 



f/ 2 r . 9 / f/A (/ (77/ \ 



Using in (7) (7'), m (2) (2'), with D constant, or in (6) (6'), 

 we find 



and 



It is to be remarked that (8) and (8') are satisfied with zj or v 

 substituted for h. 



I. Solutions for Rectangular Coordinates. 



The general type solution of (8) is h=e ax ^€ yt } where a, /3, 

 y are connected by the equation 



a2+/32= rW (9) 



For waves or oscillations we must have y=a*/— 1, where <r 

 is real. 



I a. Nodal Tesseral Oscillations. 

 For nodal oscillations of the tesseral type we must have 

 = mV — 1, £=%v — 1, where m and ?i are real; and by put- 

 ting together properly the imaginary constituents we find 



i n sin .sin sin ,„_ 



A=C <r£ m# ?zv (10) 



cos cos cos &' ' ' ' \ xyj j 



where m, ?i, cr are connected by the equation 



m +n ~w~ ( } 



Finding the corresponding values of u and r, we see what 

 the boundary-conditions must be to allow these tesseral oscil- 

 lations to exist in a sea of any shape. ~So bounding-iine 

 can be drawn at every part of which the horizontal compo- 

 nent velocity perpendicular to it is zero. Therefore to pro- 

 duce or permit oscillations of the simple harmonic type in 

 respect to form, water must be forced in and drawn out alter- 

 nately all round the boundary, or those parts of it (if not all) 

 for which the horizontal component perpendicular to it is not 



