26 Hydraulics of the Mississippi River. 
from these jesse - every tenth of depth, though never absolutely 
zero, are alw mall, and are positive in a part of the curve 
and eaediaee in ” the rest. A mathematically regular curve was 
not to be expected; but the approach toa parabola i is so near as 
to warrant the conclusion that this is the curve according to 
which the differences of velocity are regulated, and according 
to which, therefore, the resistances are distributed through the 
moving mass 
The equation of the parabola is easily deduced. In the equa- 
tion of the common parabola, if we assume particular values, 
Ly Y;,, for the codrdinates, we shall obtain the expression, 
eee tle y*; 
3 
Bigs 
and if an ee increase, =x, be given to all the abscissas, 
this will becom : 
2—2 = wll ae or ry y?+2,. 
In applying this equation to the observations, x, is to be re- 
placed by the maximum velocity, and «,, and y,, by the values of 
those eudipaies at the points most distant from the axis. The 
values of x are then to be computed for all the depths at which 
actual observations of velocity have been made. The sums of 
e computed and observed values are then to be compared, and 
their difference, if any, divided by the number of points observed, — 
is to be app lied as a aBleesys to «, and to all the values of a: 
an operation which amounts to moving the whole curve slightly | 
along Me aXis, ry altenitg its curvature. By varying - : 
depth of the axis, or the position of the point z,, y,, it m 
easily be found where the closest oe ner between the e: : 
servations and the computations can be se 
ured. : 
The equation of the grand mean curve of subsurface velocities — 
having been obtained by the processes here described, the degree _ 
of its accuracy may be tested by comparing severally the values — 
of the velocities computed by means of it, for all the points be- _ 
neath ~ ersten at which MS deen were actually observed, with : 
the ta mean observed value The comparison as ma 
Sartishes | the following results : The actual maximum velocity — 
erved being in feet 3°2611, the greatest difference found be 
tween pis Desa aa and any ‘observed mean value is onl ¥° 
and the t is ‘0006. The sum of all the differences, nkek. 
without found to sign, is only ‘0245, which is Jess than three- 
tenths of an inch. This test is certainly very satisfactory; but 
‘it, e porroborted by others to be presently mentioned. 
_ The forms of the ma hee Dregs subsurface velocities at high 4 
tages of the river, indicated 
rvature consequent upon a change of mean velo- 
d 
: 
