and their relation to the Velocities of Currents. 3 
drops succeed each other with sufficient rapidity, in short, if 
they constitute a jet of water falling into the current, the suc- 
cessive wayes will, by their intersection, give rise to a continuous 
series of points elevated above the general level of the liquid, and 
forming a ripple more visible than the waves which it envelopes. 
If we replace the jet by a solid cylinder, the effect will be essen- 
tially the same: the waves, it is true, may differ in form and 
even in the velocity with which they are propagated ; but, as 
before, the ripple will be their envelope. It is m this manner 
that the pebbles and other partially immersed bodies on the 
banks of a stream give rise to ripples whose forms, as we shall 
see, indicate in every case the velocities of the adjacent parts of 
the current. It is scarcely necessary to add that bodies moving 
on the surface of still water produce precisely similar effects ; the 
tipples caused by boats and water-fowl are examples familiar to all. 
5. The relation between the form of the ripple and the velo- 
city of the stream may be easily determined without even know- 
ing the forms of the waves of which the ripple is the envelope. 
Deferring this determination, however, to art. 10, let us first, 
for the sake of completeness, consider the following question :— 
The initial form of a wave being known, through what varia- 
tions will it pass as it floats down a stream where the velocity 
and direction of the current vary from point to point according 
to a given law? 
Let z and y be the coordinates of any point m on the sur- 
face of the stream, and let v and & be given functions of x and 
y, denoting respectively the velocity of the current at the point 
m, and the angle between its direction and the abscissa axis; 
the problem is to find the equation 
Gay Dil «vane fe si? (dh) 
of the wave at the ees of a time ¢, the sagen 
of the wave at the origin ‘of f that nee being known. 
At the time ¢ in question, the point m of the wave dbme will 
have two velocities ; one, A, in the direction of the normal mn to 
the wave, and another, v, in the direction ma determined by the 
angle amX=a. At the expiration = 
of the element of time dt, there- 
fore, the point m will arrive at the 
opposite angle m! of a small paral- 
lelogram, whose sides mn =)dt and 
ma=vdt have the directions above 
defined. If we call z' and y’ the 
coordinates of m!, and d= nmX the 
_ angle between the external normal 
B22: 
