and their relation to the Velocities of Currents. 15 
parallel to the abscissa axis. Lastly, when @> 1, the loops and 
cusps disappear, and are replaced by points of inflexion, H, K, 
which lie on the line €= — “. A somewhat clearer image of the 
ripple may be obtained by regarding it as the envelope of a circle 
whose centre M moves along one of the three curves of fig. 5 
and whose radius increases proportionally to the distance MN, 
measured along a parallel to the ordinate axis, between its centre 
and that portion X Y' X’ of the circumference of the circle whose 
concavity is turned in the same direction as that of the portion 
B’ MC’ of the curve along which the centre is moving. 
16. To return to the equations of the ripple : let us, for brevity, 
put 
sin (—d) =p, 00s (P—$) =¥3 
the equations (21), (22), (23) will then Rg 
(w—va)* + (y—pa—Roa)?— ap?" = ; 
| pla—va) + (8—v) (y—na— Roa) pert res 
v(@—va) + w(y—pa—Bda) + (l—2? + B?—2vB)a=0 
The first two equations, when solved for w and y, give 
ap+(B—v) /R 1 
Se ? | (25) 
Aw _ pe(B—v)FuvR | 
a u+6h ay * maaan go ee > | 
where 
i Pe 6 2B. a ie a eG) 
In these values of the coordinates of any point of the ripple the 
upper and lower signs correspond, and refer to the two distinct 
. branches into which the ripple divides itself. 
On eliminating 2 and y from the three equations (24), the 
result will be found to be | 
[a*— (1 —@y)?] 0°? — 2a8yRagd+(1+6?—28v)R?=0. (27) 
This is the equation, mentioned at the end of art. 14, which is 
satisfied by the values of ¢ corresponding to the cusps. 
17. When the value of ¢ is such as to make R=O, the two 
branches of the ripple meet, and from (25) the points of junction 
he in the curve represented by 
Plies ERB Ils ciel ihn ee alee 
y=ap+avd, 
which may be easily shown to be the involute of the circle formed 
by unwrapping the circle backwards from the point A, where the 
