14 T. A. Hirst on Ripples, 
If the three successive waves intersect in a point, the two chords 
will also pass through that point, and its coordinates will conse- 
quently satisfy, simultaneously, the three equations (21), (22), 
(23). The position of the cusps of the ripple at the moment 
the jet reaches A will be found, therefore, from the two 
equations which result from eliminating ¢ from the last three 
equations. These two equations will of course contain the 
angle yw which defines the position of the jet; if the latter be 
also elimimated, the resultimg equation will be that of the curve 
described by the cusps as the jet rotates. 
It will be at once observed that the elimination of w and @ 
from the above equations is equivalent to the elimination of the 
three variables sin (Wr—@), cos ar—@), and ¢ from those equa- 
tions in conjunction with 
sin? (r— d) + cos? (h—d) = 1; 
so that in the result all circular functions will disappear; that is 
to say, as the jet rotates, the cusps of the ripple will describe an 
algebraical curve. Particular cases excepted, the order of this 
curve is high; for instance, when the three velocities X, u, and v 
are equal, it reaches the eighth order ; when v vanishes, however, 
it becomes a circle (art. 20). 
If, lastly, we eliminate x and y from the equations (21), (22), 
(23), we shall obtain the relation between ¢@ and a which corre- 
sponds to the cusps at the moment under consideration, 27. e. 
when the jet reaches the position A. 
15. Before proceeding further, however, it will be useful to 
examine the locus (20) of the centres of the circular waves at the 
moment under consideration. On eliminating ¢ from the equa- 
tions (20), the equation of this locus will be found to be 
7—ab w= Ve—2—aR cos 
from which we learn that the curve undulates between the two 
lines £= +a parallel to the ordinate axis, and that the successive 
undulations are precisely similar, the length of each undulation 
being 27ra8. The form of each undulation differs according as 
£ is less than, equal to, or greater than 1, and in the following 
manner :—Firsi, when 6 <1, 7. e. when the velocity of the cur- 
rent is less than that of the jet, the curve consists of a series of 
loops, ABCDEF (fig. 5), the distances between the points 
B, D, F, where it touches the line X’F, as well as between the 
points C, EK, where it touches the lme XH, being 27a8. The 
branches BC and CD, however, are not symmetrical. As 8 
increases, the loops diminish until, secondly, when B=1, they 
become transformed into cusps B’, D', at which the tangents are 
