34 Hydraulics of the Mississippi River. 
In this equation, Uw, and d,, are both unknown. The equation 
for mean velocity in the vertical plane contains (as above stated) 
values of U at the surface and at the bottom, both of which are 
aeppees, from the equation just given, in = containing 
only the same two unknowns, Ug, and d,,."_ There will therefore 
be two = Sepa equations, involving only es unknowns, 
and from these the depth of the axis, on which the values of d,, 
depend, may be determined. 
This determination having been made for the four wind-forces 
he 
ment is directly proportioned to the wind-force, that’ it is equal 
for the same wind-force in opposite oat , but that it is a 
depression for an up-stream wind and an e vation for a down- 
stream wind. ee also appears that the ein of penta 
is independent of the mean velocity of the river. And although, 
in this Seveewientices the data for determining the effect of each 
wind-force are entirely independent of those for the others, yet 
' the results exhibit a remarkably close agreement. As a result 
of the ae ste obtain the following formula which is a general 
the depth of the axis, (denoted by d,,) /repre- 
senting ‘the force of the wind, and r the radius, or mean depth 
of the river 
d=(0'317-L0-06f)r. 
Since the process just described furnishes the means of ex- 
pressing the velocity at the surface or = any depth below it, : 
terms containing all the variables affect its value, it 
manifestly practicable to deduce a svete of gauging a river in 
which the true discharge shall be obtained from observations at 
one unvarying depth. But, as the ie be the so observed velo- 
city to the mean velocity is a varyin , no system of ‘gauging 
founded on this method of oder vation ty ’ be relied on, in which 
account is not taken of this variation. 
sg bears a ratio to the mean velocity in the vertical as 
this, 
e 
dHieretine to adopt the following symbols: 
has, apparently, two distinct values as employed above: :—it 
ting ad tao re itis he axis to the * catee: tied for the 
IS SE TREO ee OE ae GRE eR ee 
