1070 
Journal of Agricultural Research 
Vol. V, No. 23 
The curves and Table III show that except for a small range of heads 
on the 4-foot notch the discharges computed by the Francis formula are 
too small. The actual discharges, however, where the head did not 
exceed one-third of the length of the crest, did not vary much from those 
computed by the Francis formula and support the statement of Francis 
that his formula would give discharge values correct to within 2 per cent, 
provided the head does not exceed one-third the length of the crest. 
Nevertheless the fact that the curves plotted from the experimental 
data have no sudden breaks or changes of direction shows that no limit 
need be placed upon the head, provided the proper formula is used to 
compute the discharge. It also shows that the necessity of the limit on 
the application of the Francis formula was due to the mathematical 
shortcoming of the formula and not to any peculiarity inherent in the 
rectangular notch. The new formula not only gives greater accuracy 
within the range of the Francis formula but also permits the accurate 
measurement of discharges with the heads exceeding one-third the length 
of the crest. The maximum limit of the ratio of the head to the crest 
length with the new formula has not been ascertained, the greatest ratio 
experimented with being 1 to 1 with the i-foot notch. The parts of all 
the curves showing the discharges with higher heads, however, were 
quite consistent in all cases with the rest of the curves. A head of 1 foot 
was run over a 0.5-foot notch, but the results were inconclusive, as the 
discharges through the 0.5-foot notch do not follow the general formula. 
The new formula is more complicated than the Francis formula, but 
gives discharge values which are more accurate within the limits of these 
experiments, and since tables are generally consulted to determine the 
flow that is passing through a notch, the practical disadvantage of the 
new formula is largely overcome. If one is obliged to use a formula in 
the field for computing the discharge, an approximation usually is suffi¬ 
cient, and the Francis formula gives discharges sufficiently accurate for 
practical needs. 
Straight-une Formula 
As stated on page 1059, it was found, when the experimental data for 
the rectangular notches were plotted logarithmically, that a general 
straight-line formula could be deduced which, within the range of the 
experiments, would give discharge values as close to the plotted values 
as did the general formula deduced above. The equations for the straight 
lines best representing the discharges with the given heads through the 
different notches were found to be as shown in Table IV. 
