4, T. A. Hirst on Ripples, 
and the positive direction of the abscissa axis, 
z'=x+ (Xcosd+v cos ade, 
y'=y+ (Asin d +0 sin ajdt. 
But on changing ¢ into ¢+dt, the coordinates 2’, y' should 
satisfy the equation (1); hence 
y+(Asingd+vsina)dt=/[v7+(rAcosd+vecosa)dt, t+dt}. 
Developing the function on the right, neglecting powers of dé 
higher than the first, and recalling the equation (1), we have 
dy dy 
Asin d+ v sin a= yp C08 b+ cos a) + aH 
or, since dy 
1 
if fdyy® and cos 7 
a/ 1+ (Gs) Vola 
dy \? 5 di di 
ay / s = ay gy . e e 
1+ oe +vsin a v7, coset a (3) 
This is the partial differential equation whose general integral 
will include the equations of all possible waves which can be 
formed under the given conditions. The arbitrary function 
which this integral involves will in each case be determined by 
the known equation (2) of the wave which corresponds to t=O. 
With respect to this equation, however, it must be remembered 
that X varies with the nature of the displacement to which the 
waves owe their origin ; in our case it depends upon the magni- 
tude and velocity of the jet-—a fact which Weber’s experiments* 
establish beyond doubt. Apart from this variation, however, it 
follows from art. 2 that, even when the jet remains the same 
throughout, X may vary from point to point of a wave; in other 
words, it may be a function of # and y. In assuming J to be 
constant, therefore, approximate results can alone be expected. 
6. If we assume the direction of the current to be everywhere 
the same, and parallel to the abscissa axis, then a is always zero, 
and the equation (3) becomes 
dy\?__ dy , dy | 
WANT E! Se ee nC) 
The velocity v still remains a function of both zw and y; but 
without departing too much from the actual state of things in 
rectilinear streams, we may regard v as a function of y alone; 
that is to say, we may suppose the velocity of the current to 
sin o= 
* Wellenlehre, p. 183. 
