616 PROFESSOR CHRYSTAL 
CasE oF A LakE OF CoNnsTANT BREADTH, RECTANGULAR SECTION, 
BUT VARYING DEPTH. 
§ 22. Let b(x)=b, A(x)=bh(w), where b is the constant breadth, and h(x) the 
varying depth. 
Then the equations (7) and (8) may be written 
Th = gh(2)— : : : . (ae 
c=- = ; : ‘ : : (8') ; 
where u = h(x). 
In the case of a stationary oscillation, we shall have 
Eh(x) =u=P sin n(t - 7) : : : : : (9’) 
where 
2 2 
= + Fits) P=0 (10’). 
It will be useful in this case to give the displacement ¢, for a point P in the 
water not lying on the surface. If the co-ordinates of P be (x, y, 2) before and 
(c+&, y, 2+G) after disturbance, € and ¢ denoting as before the displacements of: the 
point (x, y, 0) vertically over (x, y, z), then, applying the equation of continuity to a 
small parallelopiped reaching from (#, y, z) to the surface and having sides parallel 
to the axes, we get, supposing z measured downwards, ¢ and © upwards, 
zdudy =(2—, + £)dw( +é/ex)dyl. Therefore, to the order of approximation contemplated, 
we have 
-4=-¢-28 
that is 
—6=2 (uaye)— 22 
we 
ON Oe be : . (i 
It follows that 
ee h(x) (h(x) -z) d/ P 
brio HO =9 202) 
_zh'(z) h(x) -2dP 
h(a) P dx (12). 
Hence as we pass vertically upwards we find all the particles of water oscillating in 
rectilinear orbits the inclinations of which vary gradually from tangency with the 
bottom to tangency with the free surface of the lake. 
§ 23. If we compare equations (7), (8), (9), (10) with (7’), (8’), (9’), (10’), it is at 
once obvious that mathematically there is no difference between the general case and 
the special one where the breadth of the lake is constant, its cross section rectangular, 
and the depth alone varies. We can pass from the one case to the other by obvious 
changes in the meanings of the variables and constants involved. We shall therefore in 
future confine ourselves to the more special case, of which it is easier to form a clear 
mental picture. 
