6 T. A. Hirst on Ripples, 
where c, is a function of ¢ to be determined from the initial form 
of the wave. If we suppose this initial form to be that of a circle 
with radius a around the origin, the required relation between ¢ 
and c, will result from eliminating # and y from the following 
equations, to which (7), (5a), and (8) become respectively re- 
duced: 
+y=a, 
wie=iP—N—ra=ey | «abana 
aar/ (c—v)?—d? + rAy=0. J 
From these we easily deduce 
ale—v) = EK tyisok ox!) aay eee 
by means of which (9) becomes 
yr/ 62 — ar? =ar(x—vt) + ¢,(a+ rt). 
Differentiating this according to c,, we have, corresponding to 
(€), the equation 
ee NED ee V2 
The equation of the wave at the end of the time ¢ results from 
the elimination of c, from these two equations, or from the fol- 
lowing two, to which they are equivalent : 
cy? =anr(x—vt)(a+ At) +¢,(aF Ad)’, 
a+Xt 
¢, = —ar : 
I “L—vt 
The result is clearly 
oy? + (ool) t= (asa y. f re 
which, as might have been anticipated, is the equation of a circle 
whose centre is on the abscissa axis at a distance from the origin 
equal to vi—the space described in the time ¢ by each point of 
the current—and whose radius, from being a, has become a+Xé, 
in consequence of the propagation of the wave with the constant _ 
velocity X. The upper sign in (12) is, of course, foreign to the 
present inquiry ; it refers to the propagation of the wave inwards, 
a case which is included in the differential equation (4) *. 
* It is worthy of notice that when v=0, the differential equation (3) or 
(4) becomes 
dy dy \2 
(zi) ao) (a) 
where k=dé, a relation which all parallel curves must satisfy, and which 
may be at once obtained from the definition of these curves. If 2, y, and 
k be regarded as the coordinates of a point in space, the above partial dif- 
ferential equation represents a developable surface generated by a plane 
