2 T. A. Hirst on Ripples, 
glaciers,’ by giving an exceedingly clear and simple explanation 
of the origin of ripples on the surfaces of streams. In the present 
paper I propose to pursue the subject somewhat further than 
he found it necessary to do, and to put his views into a mathe- 
matical form. 
2. When a spherical body—or a drop of water—falls upon 
the surface of still water, a system of concentric and circular 
waves are formed around the point of impact. The foremost of 
these waves generally exceeds the rest in magnitude, and being 
on that account most visible, will be referred to as the wave: its 
height and breadth, as well as the velocity > with which it 
recedes from the point of impact, all depend upon the magnitude 
of the body, and the height from which it fell. According to 
Weber*, this velocity > of propagation varies also with the time, 
or, more strictly, decreases as the radius of the circle formed by 
the wave increases: this variation, however, is admitted by Weber 
to be small+, and according to Poisson’s calculations, has no 
existence. In the present paper this possible variation of X is 
not overlooked, although, to obtain definite results capable of 
being compared with those of experiment, A is often treated as a 
constant; the error incurred by so doing being rendered less 
important by the circumstance that the waves with which we 
shall then be concerned cease, in reality, to be visible before 
their radu have reached any great magnitude. 
3. If we suppose the spherical body to fall into a current of 
water whose velocity v is everywhere the same, the particles 
forming the surface of the current will still be relatively at rest, 
and the wave will again be cireular in form, the centre of the 
circle being carried down the current whilst its radius increases 
with a velocity X, which we may assume to be the same as before. 
If the velocity and direction of the current vary from point to 
point, the circular form of the wave will be destroyed as it floats 
downwards, and the variations of form through which it will pass 
will, as Weber remarks, indicate in some measure the variations 
in the direction and velocity of the current at its several points. 
4. Let us next suppose a succession of drops to fall into the 
stream, the points of impact being fixed in space. Each drop 
will occasion a wave; and if there be no current, the several 
waves will form a system of concentric circles around the pomt 
of impact ; if there be acurrent, however, and its velocity be not 
too small, the successive waves will intersect one another, and at 
the points of intersection the water will be raised to a height ex- 
ceeding that of either of the intersecting waves. Lastly, if the 
* Wellenlehre, p. 182. Tt Ibid. p. 210. 
{ Mémoires de ? Acad. Roy. des Sciences de I’ Institut, 1816, vol. i. p. 165. 
See also Wellenlehre, p. 423. 
