10 T. A. Hirst on Ripples, 
stream, where the velocity diminishes from the centre towards 
the sides, will end abruptly as soon as it has reached a point 
where that velocity is less than.X; the pebbles on the banks of 
such a stream will produce no ripples. Ina similar manner, too, 
® may be regarded asa limit beyond which the velocity of a body 
moving through still water cannot be increased without visibly 
rippling its surface. 
12. For the sake of further illustration, let us assume, as in 
art. 6, that the direction of the current is everywhere parallel to 
the abscissa axis, and that its velocity v varies only with the di- 
stance y from this axis. The differential equation (15) of the 
ripple then becomes 
dy VP ava ie. 
whose integral, > being considered as a constant, is 
{dy Ver—e=de+C, . .. o = (19) 
where the constant C will be determined as soon as any point in 
the ripple is known. By (18) we can determine v whenever the 
form of the ripple is known, and by (19) we may find the equa- 
tion of the ripple whenever the law in the variation of the cur- 
rent’s velocity is given. 
For instance, we may determine the nature of the current, the 
ripples upon whose surface are parabolas. For in this case the 
equation of any ripple being 
yy? =2Qpa, 
we have at once 
Ch dn Sa p 
da Joey 
whence we deduce 
np? 
and conclude that, to produce parabolic ripples, the velocity of 
the current at any distance y from the axis of the parabola will 
be represented by the abscissa of a hyperbola having that axis 
for transverse axis (2X), the vertex of the parabola for centre, and 
a conjugate axis equal to the parameter of the parabola. 
Conversely, if the velocity of the current at any distance y 
from the abscissa axis satisfy the relation 
that is to say, if this velocity can be represented by the abscissa, 
corresponding to the ordinate y, of an ellipse having its centre 
at the origin, then the equation of the ripple, as given by (19), 
