18 T. A. Hirst on Ripples, 
circle by the two positions of the jet at which the divergence 
between the branches of the rippleis the same as before. Then, 
since the divergences are the same in both experiments, it fol- 
lows from art. 10 that the resultant velocities (R) and (R,) will 
be equal; so that by (31) we have 
u? +0? —2Quv cosw=u,? +0? —2uyv cos Wy, 
whence we deduce 
eed ur —u,* 
°= 8 ucos ar—u, cosy 
2nd. Let the positions A and A! be found at which the angle 
between the branches of the ripple is bisected by the production 
of the radius. In this case the resultant velocity (R) will coin- 
cide in direction with the radius at each of the points A and A’, 
whose angular distance asunder is 2, and from fig. 4 we deduce 
at once the relation 
ean 
= C08 Ae 
20. The case alluded to in art. 13, where the jet describes a 
circle in still water, may now be easily disposed of by making 
v=0, and therefore 8=0, in the general case treated im art. 14. 
In this manner the equations (24) become 
(av)? + (yay)? —a2’g? =0, 
(c—av)u—(y—ap)v +a2a =0,} (32) 
(w—av)v + (y—ap)u+ (1-2? )a=0; 
or, by simplifying, 
a? + y* —2a(av + yp) + a? — a°a?g?=0, } 
xp— yv +a°ad =0, . . (33) 
rv+yp—oa =0; J | 
of which the first two represent the ripple, and all three are 
satisfied by the coordinates of its cusp. If, be the value of 
which corresponds to this cusp, we find immediately, by elimi- 
nating z and y from (82), 
pW ae i et 
which will always be real and positive when wu exceeds A. This 
same value of ¢ might be obtained from (27) by putting B=0; 
by means of it and the last of equations (33) the first becomes 
which is the equation of the circle described by the cusp as the 
