94 Transactions.—JMiscellaneous. 
velocity of the moving water. The water near the surface is urged towards 
the left bank with greater force than that near the bottom. To ascertain 
the kind of motion caused by these unequal forces :—Suppose the stream to 
be straight, of uniform cross-section, to be everywhere of the same depth, 
and imagine it to be divided into a great number of layers parallel to the 
surface, each moving with different velocity. "Then, the increase of pressure 
against the left bank, due to the earth's rotation, equals the sum of the 
deflecting forces, which sum is the same as if the mean deflecting force 
acted on every layer. Therefore, the water-level at the left bank is raised 
to the same height, and the surface-line of the cross-section is inclined to 
the horizon at the same angle as if the mean deflecting force acted on every 
layer, and the tangent of the inclination is the latter force divided by the 
force of gravitation. 
The aecompanying cut shows 
the eross-section of a stream, the 
angle A being very much exag- 
gerated. : 
Let a be a small cube of water in any part of the cross-section whose 
volume is da dy dz. 
h,h, the depths below the surface of the centres of the left and right 
sides of the cube respectively. 
Jf, the deflecting forces acting on the cube a and on a similar cube in 
the middle layer respectively. 
w, the weight of an unit of volume of water. 
A, the angle of inclination of the surface of the stream from left to right. 
F, the resultant force acting on the cube a. 
Then F = w (h—A, ) dy dz—f 
= tan A. w dz dy dz—f 
os 
Er pn A n dx dy dz 
Therefore F = f,—f - 
That is, the resultant force acting on a particle in any part of the cross- 
seetion is the difference of the deflecting forces acting on that particle and 
a particle in the middle layer. This quantity is of different sigg for 
partieles situated above and below the middle line, showing that the. 
resultant force acts in opposite directions, above and below that line. 
These forces must evidently cause a circulation, as shown by the arrows in 
the figure, which motion will be combined with that down the stream, so 
that the actual motion of any particle is inclined at à very small angle to 
the direction of the channel. It is clear that a very slow motion of the 
bottom layer from left to right must cause a transfer to the right side of the 
