94 Transactions. — Miscellaneous. 



velocity of tlie 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 accompanying cut shows 

 the cross-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 dx chj dz. 



h,h^ the depths below the surface of the centres of the left and right 

 sides of the cube respectively. 



f,fi 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 = iv (h—hj dij dz—f 



= tan A . IV dx dy dz — -f 



f 



And tan A = — f^^ — r 



w dx ay dz 



Therefore F=f^—f 

 That is, the resultant force acting on a particle in any part of the cross- 

 section is the difference of the deflecting forces acting on that particle and 

 a particle in the middle layer. This quantity is of different sign for 

 particles 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 combmed with that down the stream, so 

 that the actual motion of any particle is inclined at a 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 



