and their relation to the Velocities of Currents. 13 
Let the centre of the circle be taken as the origin of coordi- 
nate axes, one of which—the ordinate axis—is parallel to the 
direction of the current. To fix our ideas, let us suppose, too, 
that after having rotated for an indefinite period in the direction 
opposed to that of the hands of a clock, the jet has at length 
reached the position A (fig. 3) defined by the angle AO X=wW. 
The centre of the circular wave which the jet originated when in 
any position B, will have been carried with the current along 
the lme BC parallel to the ordinate axis, and its radius, from 
being zero, will have increased to a certaim magnitude C M. 
The arc A B, the line BC, and the radius CM being described 
in the same time, we shall clearly have the proportions 
AB: BC:CM=u:0:); 
so that, on representing the angle AOB by @, and the ratios 
= and ~ by « and # respectively, we shall have 
AB=ad, CM=aa¢d, BC=aBd; | 
and the coordinates &, » of the centre C of the circular wave 
will be a “8 
eel a ae t lea ptee hi) 
n=asin (—¢) + aB¢ ; 
whilst the equation of this wave will be 
[x —acos (y—$)]? + [y—asin (y—$) —a8o]?—a°a2g?=0. (21) 
The equation of the immediately succeeding wave will be ob- 
tained from this by changing ¢ into 6+d¢; the intersections 
of the two waves will be two points on the ripple, and their co- 
ordinates will clearly satisfy the equation (21), as well as its dif- 
ferential according to ¢. The equation of the ripple, therefore, 
at the moment the jet reaches the point A defined by the angle 
ar, will result from the elimination of ¢ between (21) and the 
equation 
[v—a cos (Yr—¢)]sin (—¢) —[y—asin (b}—¢) —a8 4] 
[cos p—)—B]+axe*M=0, . . . . « . (22) 
which is simply that of the chord of intersection of the two suc- 
cessive circular waves. In a similar manner, the equation of the 
chord of intersection of the second of the above waves, and a 
third, immediately following the same, will be found by putting 
o+d¢ in place of d in (22); and the coordinates of the point 
in which these two chords cut each other will satisfy both (22) 
and its differential according to @, which is 
[v—a cos (f—¢) ]eos (f—¢) + [y—a sin (—¢)— 489] 
sin (Wr —¢)+a[l1—a?+4 6?—28 cos (w—g]=0. . (23) 
