12 T. A. Hirst on Ripples, 
to point according to imperfectly known laws. Now of all con- 
ceivable motions which might be imparted to the jet, a circular 
one is beyond doubt most feasible ; so that we are naturally led 
to inquire what will be the form of the ripple produced by a jet - 
which, as it falls into still water, describes a circle with a given 
constant velocity u. 
We shall throughout assume the velocity with which the 
waves are propagated to be independent of the magnitude of the 
latter, and, in accordance with art. 4, we shall seek the envelope 
of the several waves which the moving jet originates. There is 
one case where the nature of this envelope can be at once deter- 
mined: it is when the jet moves with the same velocity » as the 
waves. For at the moment when the jet arrives at a point A 
(fig. 2), the wave which it produced when at B will have acquired 
a radius equal to the arc AB; instead of intersecting, therefore, 
every two successive waves around B and B! will touch each 
other, the difference between their radi being equal to the 
distance B B! between their centres, and their point of contact 
A! will be in the tangent to the circle at the point B; in fact, 
the ripple in this case will be the involute of the circle A A! C, 
or the curve formed by unwrapping, under tension, a string ori- 
ginally wrapped round the circle as far as A. If the velocity of 
the jet were less than 2, the successive waves would precede the 
jet, and neither intersect nor touch each other; in other words, 
there would be no ripple. But if the velocity of the jet exceed 
r, the ripple will separate into two branches (fig. 7), one of which 
will be outside the circle, whilst the other will enter it; but 
since the waves continue to increase in radius, this latter branch 
will necessarily leave the circle again, and never re-enter it. In 
art. 20 we shall find, in fact, that this branch, after approaching 
to within a certain distance of the centre, suddenly turns and 
recedes, the turning-point or cusp C bemg due to the intersec- 
tion in that point of three successive waves. In experiments this 
cusp 1s tolerably well defined; and its position is the more im- 
portant, since it bears a very simple relation to the velocities wu 
and 2; in fact, we shall find that the distance of the cusp from the 
centre of the circle described by the jet has to the radius of that 
circle the same ratio that the velocity \ of propagation has to the 
velocity u with which the jet moves. 
14, For the sake of future applications, it will be more con- 
venient to deduce the form of the ripple above considered from 
the solution of the following more general problem. 
A jet which describes, with uniform velocity wu, a fixed circle 
with radius a, falls into a current whose velocity v and direction 
are the same at every point of its surface; required the form of 
the ripple. 
