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 mathematical 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 λ 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[1], this velocity λ 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[2], and according to Poisson's calculations, has no existence[3]. In the present paper this possible variation of is not overlooked, although, to obtain definite results capable of being compared with those of experiment, λ, 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 radii 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 circular in form, the centre of the circle being carried down the current whilst its radius increases with a velocity λ, 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 point of impact; if there be a current, 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 exceeding that of either of the intersecting waves. Lastly, if the
