94 
HARRIS. 
most continuous bodies are not as yet recognized in analysis. It is there¬ 
fore often necessary to fall back on methods of approximation, referring 
the proposed system to some other of a character more amenable to 
analysis, and calculating corrections depending on the supposition that 
the difference between the two systems is small.” 
Let ns consider first the problem of free oscillations for 
long-wave motion in a body, of uniform depth. The state¬ 
ment is simple enough. We have only to find a suitable 
solution of the partial differential equation 
d'C 
such that = 0 at every point of the rigid vertical bound¬ 
ary. C denotes the vertical displacement of the surface, 
tc 2 — gh, and v refers to a normal to the boundary. The 
problem of the vibration of a stretched membrane differs 
boundary. Circular functions in x, y are suitable for rectan¬ 
gular or square areas. For a simple mode of oscillation the 
number of terms in the solution is small—-one or two. The 
same is true for certain triangular areas. Bessel’s functions 
are appropriate for circular areas; but here I will digress 
long enough to ask one question. Consider the slowest 
mode of oscillation (symmetrical about the center) of a square, 
an hexagon, and a circle, all having the same period. The 
solutions may be written : 
cos lid cos —7= lx cos -w ly, 
V 2 V 2 
where l/c — the “ speed ” or 2r -f- period and w = \/ x 2 -f y 2 . 
Can this last solution be approached through solutions for 
