vg ie 
Sa ne 
2 
Report of Messrs. Humphreys and Abbot. 199 
In order to determine the nature of the function 9(z) and the 
constants which must enter into it, a collection was made o 
the available data which had been furnished by the survey, or 
which could be gleaned from the publications of other observers; 
in magnitude from the dimensions of the Mississippi at high 
water, down to those of a small canal. In regard to slope, it 
was to be considered that a portion is expended in overcoming 
the irregularities of the channel and the changes of cross-section : 
and a portion, in compensating for the loss of living force at 
bends. It is only what remains after these effects have been 
subtracted, which constitutes the equivalent of the resistance 
of a straight and regular channel. The effect of bends must 
provided for in an independent formula; and the amount of 
vided for in the modification which they introduce into the con- 
stants which are derived from observation, on the supposition 
that, after bends have been allowed for, the channel is straight 
and regular, and the movement in it uniform. The method 
pursued by most writers, of putting 9(v)=Av+Bv?, and then 
seeking values for the indeterminate coefficients which shall 
most nearly represent the observations, was tried by the authors 
of the report, making 
F, 
rs Az+-Bz?, or LoA+By, 
in- which “ and z are co-ordinates in the equation of a straight 
line; but they found that a straight line would not represent the 
observations, and that the involution of z produced expressions 
of troublesome complexity. They then put . 
r,3 
TS—Use", or et 
and plotted the values of C as ordinates to r,s,andv. The 
plots with r, and w produced irregular curves following no appa 
rent law. That with s was quite regular. It was inferred 
therefore that C is some function of the slope. After a very 
long series of trials, with a view to discover this function, the 
: 8 
Cai55 
was adopted, as most satisfactorily fulfilling the tequired con- 
ditions." Sahetitnting this, therefore, in the formula, it becomes 
