L. Nickerson on'the Periodic action of Water. 155 
from that point, the. difference a,—a, should show the flactua- 
tion of height due to a periodic change in the discharge. Now 
2 
when sin a = ao it is also by the law of the formula =sine’ 
_ of that transverse section of which the second member shows the 
‘resistance ; therefore 
Sine = sina’, 
therefore the velocity of the pool = the velocity of the stream, 
_ and the surface line of the pool is a parallel line with the bed ; 
or @,—a,=0, therefore it is circumstanced as in the original 
stream. There is no backwater, no difference of pressure and no 
vibration. : 
sineé=Q0 | 
the surface becomes level, for there is no velocity, no flowage, 
therefore no resistance, z — =(0; again a,—a,=0; and there 
18 NO action, periodic or otherwise. ode 
So we see that there are two points at which the vibrations 
cease: namely, when the water is sufficiently high to flow ove 
the dam without much remou, as with a stream undammed, 
and with its surface a line nearly corresponding with the surface 
of the original stream; and again when the water is so low as 
_ tomake the difference between the hydrostatic and hydraulic 
_ Pressures very small. Of course these limits are much circum- 
scribed by the inertia of a large body of water which has con- 
-Stautly a tendency to absorb and soften these vibrations. The 
__ Mnost violent palpitation should then occur when 
2 ed ee? oh 
| sina =(<5,.5,) 4 | 
_m being a new quantity to be found by a knowledge of the 
Stream, 
Again, if we put 2.2 =1, or = 1 or when the height 
a 2 
° the velocity of the original stream becomes equal to one-half 
¢ depth of the same, we have 
t, tending to rise infinitely, is checked by the action of gravity, 
falls back peat ne plete stream, and tends to form a 
5 . 
