and their relation to the Velocities of Currents. < 
drawing BC parallel to M M’, we have, as before, 
! 
as = sin B/BC= sin 0= ~ 
where @ is the angle between the directions of the ripple and the 
current at the point M. The velocity v being constant, @ and » 
will decrease simultaneously ; so that, according to Weber, the 
ripple should consist of two curved lines, AM, AM,, turning 
their concavities towards each other. This property of the wave 
suggests a crucial experiment as to the variation of 1; to apply 
it, however, a perfectly uniform current would be required, or 
what is equally difficult to realize, a jet must be made to de- 
scribe a right line with perfectly uniform velocity over still water. 
In ordinary experiments, as will be hereafter seen, the ripple, as 
long as it remains visible, and as far as the eye can judge, is 
rectilinear. 
Again, it may easily be shown that when the velocity and 
direction of the current vary from point to point, so as to destroy 
the circular form of the waves, the law of art. 9 is still fulfilled. 
In fact, in the immediate vicinity of the pot M of the ripple 
we may regard this velocity and direction as constant; and in 
place of the non-circular wave, to whose intersection with the 
immediately preceding and succeeding waves the ripple at M is 
due, we may substitute a circular one osculating the real wave 
in M, increasing with the same velocity X, and moving parallel 
to the current at M with the velocity v which exists at that 
point. This fictitious circular wave will clearly produce a ripple 
coincident with the actual one in the neighbourhood of the point 
M, and thus the relation (16) between XA, v, and @ will still 
exist at that point. 
11. From the relation 
sin 0= Ms 
v 
7 
2 
when the velocity of the current is equal to the velocity with which 
the wave is propagated, the several waves all touch a line at 
right angles to the direction of the current; they will con- 
sequently touch each other, and the ripple will become reduced 
to their point of contact. When A exceeds v, 0 is imaginary; im 
fact, in this case the waves will clearly be propagated up the 
stream, and will no longer intersect. Strictly speaking, however, 
this is the case only when the waves are produced by a discon- 
tinuous series of drops ; experiment shows that when a solid cy- 
linder or jet is partially immersed in so slow a current, the water 
flows past it without its surface suffering any visible*disturbance. 
From this it follows that a ripple which has been produced in a 
it follows that when A and v are equal, O= =; that isto say, 
