206 Hydraulics of the Mississippi River. 
The old formule all give here a velocity largely in excess ; 
whereas in large streams they are almost invariably in deficiency. 
The new formula represents these cases with as close an approac 
to observation as any others. The explanation of the anomaly 
is not obvious. The example of nearest general agreement of 
results, appears to be the small river Haine, No. 20, which gives 
the following :— 
Vel.observed, - ~ - 24947 Prony-Weisbach, - - - 26414 
Chezy-Young, - - - 24046 oung, - - - 2°8893 
Dubuat, - . - - 24494 St. Venant, - - - - 95540 
Girard, - - - . 32749 Ellet, . . - - 19707 
Prony-Eytelwein,- - - 25987 Humphreys and Abbot,- - 24690 
If we examine the numerical ratio between the sums of the 
errors of the several formule in these thirty cases taken without 
sign, as given in the table, to the sum of the observed 
velocities (115°4847), we shall find it to vary from twenty-two 
rcent for the formula ef Dupuit, to thirty-nine per cent for 
that of Ellet. The formula of Humphreys and Abbot gives 
five and a half per cent. If we take the algebraic sum of these 
errors, this last ratio is reduced to three per cent; which is the 
tendency, as shown by this table toward excess. Examining 
the other formulz in the same way, we shall see that they are 
all in deficiency, with the exception of Girard, who leans on 
the side of excess to the extent of eleven and a half per cent. 
The Chezy-Eytelwein formula gives a ratio of twenty-five per 
cent when the arithmetical sum of the errors is compared with 
the sum of the velocities; the Chezy-Downing formula gives 
twenty-three per cent on the same comparison. In these 
the algebraic sum of the errors shows a tendency to deficiency 
of fifteen and a half per cent for the first, and nine and a 
for the second. Of all the old formule, the Dupuit appears to 
be the best; for the arithmetical sum of its errors bears the least 
ratio of all of them to the sum of the velocities; and the oppo- 
site errors, in these examples at least, almost exactly balance. 
The second method employed by the authors of the report, to 
test the accuracy of their formulz, consisted in computing the 
differences of level between points of the river distant from each 
other, in regard to which this difference had been ascertained by 
measurement. The same computation was made by Mr. Hllets 
formula also, the results being introduced into the table along 
with those derived from the new formule, for the purpose 
comparison. No computations were made in this case from the 
other formule, their large errors already showing their inappl 
eability to natural streams. An exception was made in favor 
of Mr. Ellet’s, because it had been expressly designed for rivers. 
-he present test applies equally to the bend formula, and to that 
for mean velocity. The following table embraces both data and 
