16 T. A. Hirst on Ripples, 
jet has reached. Further, if be the angle, opposite to the side 
Xr, in a triangle whose sides are respectively proportional to the 
velocities X, u, and v, we shall have the relation ; 
Mau2+v2—Qweoskh; . . . . (29) 
or, dividing by wu? and introducing the ratios « and @ (art. 14), 
a? =1+ B?—2B8 cos a 
This, compared with (26), shows that the condition R=O is 
satisfied when 
y=cosr, and w=+sind; . . . (30) 
so that the result of eliminating w, v, and @ from (28) and the 
equation R=O will be 
zcosh+y sin A=. 
This is the equation of the curve described by the junction points 
of the branches of the ripple as the jet describes its circle, which 
curve, as is at once seen, consists of two right lines touching 
that circle at points B and B’ (fig. 6), whose angular distances 
BOX, B’O X from the abscissa axis are each equal to a, Now, 
since an angle x fulfilling the condition (29) can always be found 
when the velocities A, wu, v form a triangle, that is to say, when 
any two of these velocities together exceed the third, we conclude 
that under these conditions the ripple will always break up into 
closed curves or loops, in consequence of the two branches of the 
ripple, which in other cases are always distinct, meeting each 
other. These meeting points describe the tangents (30) BC, B/C’ 
as the jet rotates, and their position at any moment is determined 
by the intersections C, C’, &c. with these tangents, of the involute 
ACC’ of the circle,—the latter being supposed to move with 
the jet. 
The physical character of these meeting points, as will be im- 
mediately shown, is that the ripple is there least prominent, so 
that the tangents (30) BC, B! C’ represent two lines along which 
the surface of the current is apparently least disturbed. If the 
angle between them, which is equal to 2\, were determined by 
experiment, and the velocity u of rotation were also known, the 
equation (29) would serve to determine either of the velocities 
» or v as soon as the other was given. 
18. The above results will be further elucidated by consider- 
ing the resultant relative velocity of the current and jet at any 
point A, fig. 4. This resultant will clearly be represented in 
direction and magnitude by the diagonal A R of a parallelogram 
whose sides A V parallel to O Y, and A U perpendicular to A O, 
respectively represent, in direction and magnitude, the velocities 
