and their relation to the Velocities of Currents. 19 
jet rotates. Putting r in place of Va?+y?, it leads to the 
relation 
Oe 5 MS ST ale et Je SEND 
expressed in art. 18. 
To obtain simple expressions for the coordinates 2,, y, of the 
cusp corresponding to a given position of the jet, it is merely 
necessary to observe that ie form of the present ripple does not 
alter as the jet rotates; so that we may refer its equation to co- 
ordinate axes which rotate with the jet,—the abscissa axis being 
at the angular distance AOX=d, behind the jet (fig. 7). By 
so doing w sand v become respectively sin (f; —¢) and cos(¢,— od); 
and the two last of the equations (33) give, on substituting for co) 
its value (34), 
yaar lab, t Ce aes Ge 
w= ea; 
so that if we call 0, the angular distance AOC of the cusp C 
behind the jet, we have, in virtue of (35), and sinceCOX=¢,—4,, 
sin (6—0,)=\/1—2?, cos ($,—9,) =e, 
ee i Me agg 24 
whence it follows that @, increases as « diminishes, in other words, 
as w Increases. 
Again, since the berigent to the ripple at any point coincides 
with the tangent to the circular wave which there touches it, we 
have from the first of equations (82), 
dy u—av 
pe Lape cante weet, he rome (38) 
With reference to our new coordinate axes, the values of y and uw 
' at the cusp are 1 and C respectively ; and these, together with 
the values in (86), give, when substituted in (38), 
=) KS aA—2, a MME Sea 
a Yy a 2, 
and show that the radius vector OC is the tangent at the cusp. 
The polar equation of the ripple, with reference to an axis 
passing through the jet and moving with it, is easily seen to be 
r?— ar cos (b— 0) =a7a*d?— a’, 
28 9 » e (39) 
r sin (b— 0) =2a7ad, 
C2 
