Report of Messrs. Humphreys and Abbot. 35 
-D==depth of river. d=distance’ below the surface (variable). 
dj—=depth of axis (line of maximum velocity in the eonage aaah 
m= depth of line of mean velocity in the vertical plan 
==velocity at any point in the vertical plane. 
Va,=velocity at the axis, or maximum velocity 
V,,==mean velocity in the vertical plan 
Vv, =velocity at the surface. Tyseee at the bottom. 
Then V,,D will be the value of an area equal to the t 
rectangles and two parabolic segments concerned in the ae 
mination of the mean velocity. The truth of the following 
equation is therefore manifest :— 
V.D=&(Ve,—Vo)4,+Vod-+2(Va,—Vo) (D—d,)}+-Vi(D—d)) 
Which, reduced, becomes, 
VetVet Vet 3e(V, —V>). 
tees now the general equation for the vertical plane, heretofore 
en, 
* 4 4/d—d,\? 
V=Va,— (bv) "d,2=Va,— (bv) = ‘) 
nate substitute in it the value of d at the surface, =e ee at the 
m, =D, and we have the two expressions follo 
2 
Vv o=Va—(o)' (7) rn oh a4) 
These values of V, and V, being introduced into the expres- 
sion for the value of Vm, we shail have, after reduction and 
yonnnan 10n, 
Hj4 Hed) 
Va=VeH(0o)" (t+ 75). 
And eo ealties this value of Vg, in aie foregoing § eral 
peopel for velocity in the vertical plane, there is t after 
uctio 
aie = 92 
sori te 
Divide the identical equation, V,,=V,,, member for member, 
by this expression, and we obtain finally. ‘the general ratio of the 
velocity at any depth to the mean velocity in the vertical plane, 
¥. P* 
Vv ale oh a 3d,D —3d? 6dd,) 
3D2 i 
in which d, must be replaced by its value as deduced from the 
inyestigation of the effect of wind-force, in order that all the 
variab es may explicitly appear. 
