and their relation to the Velocities of Currents. 7 
9. Let us next consider the ripple which envelopes a system 
of waves having at their origin the same position and form. If, 
as in art. 5, 
y=fle,t) . - (1) 
represent the equation of any wave, ‘ 
y=flz, t+ dt)=flz, t)+ 2 dt + &c... 
will be that of the next preceding wave, and the values of # and 
y which satisfy both equations, or which satisfy (1) and the 
equation 
dy 
= 8 ty RRM CS Ore a) EO 
Yo, (13) 
will refer to a point on the required ripple. In short, the equa- 
tion of this ripple will be the result of the elimination of ¢ from 
the equations (1) and (13). If we differentiate (1), regarding ¢ 
as a function of 2 determined by (13), and use brackets to di- 
stinguish partial from Sone differential coefficients, we have 
oH _(@) 4 cee shi 
da 
1) ot, 
bag ae ete e 
an equation which merely expresses the well-known fact, that at 
their point of contact the wave and ripple have the same tangent. 
But it was shown in art. 5 that the equation (1) satisfies the 
partial differential equation (3); and the latter, by means of (13) 
and (14), becomes transformed into the ordinary differential 
equation 
hence by (13), 
— | 
ry 14 E+ vsinaxol cose, Say Ae cA 
which is clearly that of the ripple. If the coordinate axes be 
turned around the origin until the abscissa axis 1s parallel to the 
direction of the current at any point M(ay) of the ripple, then, 
since «=O at that point, (15) becomes 
which is constantly inclined at an angle of 45° to the plane (xy) or axis k, 
and the sections of this surface made by planes parallel to (wy) will consti- 
tute a system of parallel curves. If the base of this system be a curve 
traced on the coordinate plane of (xy), the generating planes of the deve- 
lopable will always touch the same. The developable, in fact, has for its 
edge of regression a curve which cuts at an angle of 45° all the generators 
of a right cylinder whose base is the evolute of the curve traced on the 
plane (zy). 
