Report of Messrs. Humphreys and Abbot. 33 
the corrected discharge. The truth of the following proposition 
will then be manifest :—As the recorded discharge is to the cor- 
rected discharge, so is the observed mean velocity, five feet below 
the surface, to the velocity which would have been observed at 
the same depth had it been calm. But, had it been calm, the 
axis would have been at the depth ‘317. In the general equa- 
tion of velocities, therefore, 
=v,—(n)ta,2 + 
if d,, be put for the distance from the known position of the axi 
to the point five feet below the surface, and the velocity ee 
for that point by the last proportion be put for V, there will 
remain only one unknown quantity, which is Vg, or the maxi- 
mum velocity in the vertical curve. * This velocity is therefore 
easily deduced, and, being substituted in the same equation, will 
enable us to compute the mean velocity in the entire vertical 
plane. For this mean velocity, being derived from the areas of 
the rectangles and parabolic segments, above and below the axis, 
which form the figure bounded by the curve of velocities at one 
end and by a vertical line at the other, the extreme length being 
the maximum velocity in the plane, and its breadth the depth 
of the river, is determined when the velocities at surface and 
bottom are given along with the maximum velocity and depth. 
The last named velocity is that which was just found; and, by 
the help of this, the equation gives the other two, when’ the 
r values of , viz., distance from the axis to ‘the surface 
and distance from the axis to the bottom, are substituted. 
his operation was performed for each of the wind- -forces, 1, 
2,3, and 4. In each case, the unbalanced wind-force of the 
olseteationd was but a fraction of the total force of the wind at 
the given intensity. It was desirable to know the effect due to 
ee entire force; and, in ae to arrive at this, it was only 
to reverse the o operation. Thus, taking the corrected 
apprcsiinate discharge as the discharge due to a calm, and in- 
ng and diminishing it by the amountof the empirical cor- 
rection corresponding to each wind-force successively, we may 
obtain the hypothetical discharge due to the full wind-force in 
pate down or up the stream. Then the proportion may be 
ted:—As the corrected discharge is to the hypothetical dis- 
ee ge, so is the velocity in calm, five feet below the surface, to 
the Pelocity at the same point under the assumed wind _-force. 
The value so determined may be substituted for V as before, or 
for U on the left of the more general equation following; which 
differs from the former only in replacing V, the velocity in a par- 
ticular plane, by U which is intended to denote aa Bie in 
the mean of all planes parallel to the axis of the ri 
UU z,—(b0)*d,2=Uy—(0" bkasthess: 
Am. Jour. Sc1.—Szconp Sexies, Vou. XXXVI, No. 106.—Juty, 1863, 
i] 
