and their relation to the Velocities of Currents. 5 
remain the same at the same distance from its banks, and to 
_ vary only on crossing the stream. Under these conditions the 
equation (4) is integrable; and if X be constant, one of its com- 
plete integrals will be found to be 
fdy/(c—vP?—N=AMw—ct)+ O(c); . « (5) 
where ¢ is an arbitrary constant, and ® an arbitrary function. 
The general integral will result from the elimination of c between 
this equation and its differential according to c, which is 
ae =35 t Meets Hage) se yee nie) 
7. The arbitrary function ® may be determined from the 
known equation 
Ie. = One re wae Fe ee ai ee anen) 
of the initial wave in the followmg manner. Putting ‘=O in 
the equations (5) and (6), we know that the result of eliminating 
c from the equations 
fdy/(e—v)? —N =a +(e), . . . « (5a) 
(c—v)dy ae 
rarer Diy ake ns cota poet (OO) 
must coincide with the equation (7). But eliminating c from 
these equations is equivalent to replacing c in the first by a func- 
tion of x and y determined from the second. Now on differen- 
tiating (5a) on the hypothesis that c is a function of 2 and y thus 
determined, we have, in virtue of (6a), 
dy V (ec—v)?—-VN=X aa, 
an equation which ought also to coincide with the result of dif- 
ferentiating (7), that is, with 
d¥ d¥ 
a tae ae —— dz=0, 
We conclude then that 
eee 
ap V (e—2)? SH eae ee a giv at Behe yay (SB) 
The result of eliminating x and y from the equations (7), (5a), 
and (8), therefore, will lead to the required relation between ®(c) 
and c. 
_ 8. As an example, let the velocity v of the current be constant, 
or the same at all points. The complete integral (5) of the 
equation (4) will then become 
V (e—v)?—AN=Aa—ct) +c, spe) ete 46 hia} 
