and their relation to the Velocities of Currents. 17 
of the current and jet. The angle at U being equal to the angle 
AOX=7, the magnitude of this resultant is expressed by 
(R)= Vu?-+v?—2uveosp, . . . . (381) 
and becomes equal to A, the velocity with which circular waves 
are propagated, when w=), that is to say, when the jet reaches 
either of the points B, B’ of the foregoing article (fig. 6). For 
all positions of the jet in the are B X Bb! the resultant (R) will be 
less than 2, and for all points in the arc B X! B’ greater; so that 
by art. 11, the jet will only commence producing a ripple when 
it reaches the point B, and will cease doing so as soon as it 
reaches the point B’. At these points, B and BY, the immediately 
adjacent circular waves of the incipient ripple touch each other,and 
the point of contact is very slightly elevated above the surface of 
the current at adjacent points (art. 11); further, since the centres 
of these waves proceed with the same uniform velocity down the 
eurrent, whilst their radii increase with the same velocity, they will 
clearly continue to touch each other along the line BC or B'C’; 
so that these lines, as above remarked, will be lines of least appa- 
rent disturbance. They do not in general touch the ripple; im fact, 
they do so only when the resultant velocity (R) =) coimcides in 
direction with that of the radius OB (or O B’), in other words, 
when v?=)?+ w?, as may be seen byaglance at fig. 4. This case 
of contact is represented in fig. 6, where A DCE represents the 
first closed loop of the ripple when the ject has reached the poimt 
A, and B!D,C, E, what this loop becomes when the jet reaches 
the point B’. The curve D'C' Hi! represents a portion of the 
second loop of the ripple corresponding to the position A; it is, 
in fact, a portion of what B’D,C,E, becomes as it floats down 
the current during the time that the jet is describing the 
are BBA. 
19. The law enunciated in art. 9 leads to a simple method of 
determining the velocity of a current ; to apply it, however, the 
velocity X must be first determined, and the velocity cf the cur- 
rent experimented upon must be greater than A. The formula 
(31), however, suggests two other methods of determining the 
velocity of a current, which do not require a previous determina- 
tion of X, and may be applied to all currents. They are briefly 
the following :— 
Ist. The jet having a known velccity uw, let the positions A 
and A! (fig. 4) be found at which the divergence between the 
branches of the ripple has a given magnitude, and call 2W the 
angle between the radii OA and OA!. The line bisecting this 
angle will be perpendicular to the direction of the current. Let 
the experiment be now repeated with a different velocity u,, and 
let 2, represent the angle now subtended at the centre of the 
Phil, Mag. 8. 4. Vol. 21. No. 137, Jan. 1861. C 
