and their relation to the Velocities of Currents. 1] 
would be 
J dyp/ @—2— Fy =a +C, 
where c= 53 on integrating, we find the equation of the ripple 
in this case to be 
ENE, 
cy V a? —n?— cy? + (a? —X2) are sin 
7 nile G22 
where the arbitrary constant has vanished on assuming that the 
ripple passes through the orig. The tangent of the inclination 
of the ripple to the current at any point 2, y has of course the value 
dy _ r 
dx J @— rn? — Py? 
which varies between the minimum limit a the origin, 
a — 
and the maximum o at the point A, whose coordinates are 
ah az—r2 a a Jer? 
Cc 
4 cr 
Further, since the equation is unchanged when 2 and y are 
replaced by —z and —y, it is evident that the ripple consists 
of two similar branches in opposite quadrants, and has a point of 
inflexion at the origin, where it is least inclined to the current. 
‘Proceeding from this point, its inclination to the current in- 
creases until, at the points A and A’, current and ripple are at 
right angles to each other. Here the latter ends abruptly, in 
consequence of the velocity of the current having become equal 
to that with which the wave is propagated (art. 11). 
In Plate I. fig. 1, let O X be the direction of the current, and 
let its velocity at any point of a line parallel to O X be repre- 
sented by half the chord which is intercepted upon that line by 
an ellipse, X Y X’ Y'. Let the semi-chords A B and A! B! repre- 
sent the velocity ) with which the wave is propagated. The 
curve A’O A will then represent the ripple produced by partially 
immersing a body at any one of its points, as A’. The tangent 
OC at the point of inflexion O is easily constructed, since it 
passes at the distance X= XC=AB from the vertex X*, 
13. In order to test by experiment the law enunciated in 
art. 9, two methods suggest themselves. First, to examine the 
ripples produced by a stationary jet falling into currents of dif- 
ferent but known velocities ; and secondly, to give to the jet or 
partially immersed body a definite motion, the water being mo- 
tionless. This second method has many advantages, arising 
from the fact that the motion of the jet is more under our con- 
trol than that of a current, which in general varies from point 
* Compare the fig. in chapter 29 of ‘ Glaciers of the Alps.’ 
