24 The Colorado Experiment Station. 
Francis and experiments have continued at Lowell. While the 
subsequent experiments have not been published, a personal exam¬ 
ination of the appliances seems to show that there might be much 
of great value. Tests were made by Mr. Francis of the flow over 
weirs and the water was measured in a large basin containing some 
12,000 cubic feet. By taking account of the contraction at the 
sides and at the crest of the weir, Francis succeeded better in 
producing a formula which represented the discharge. 
The equation assumed and agreeing closely with his experi¬ 
ments is given with the form. 
Q=aLH% 
Where Q is the quantity of water in cubic feet per second, L, the 
effective length of the weir in feet. It is not necessarily the same as 
the actual length of the weir. It is mentioned more fully on the next 
page. 
a is a coefficient supposed to be constant. The limitations in length 
and depth state conditions within which it is practically constant. 
H, the depth of water flowing over the weir, in feet. Because of 
the contraction, this must be measured far enough from the weir to be 
free from its influence. If the water approaches with a current, this depth 
needs to be increased by a correction indicated by theory. The object is to 
give the equivalent depth if the water came from rest. This correction is 
troublesome to make. In practice it is better to so check the velocity of 
the current that the correction will be so small that it may be neglected. 
Table H in the appendix gives the amount of this correction which is 
needed to multiply the results obtained by the indicated operations in the 
measured quantities, in order to give Q the discharge. 
From his experiments, an abstract of which cannot convey 
an idea of the care and skill used in the experimentation, Francis 
adopted the value of 3.33 for a; 3would agree a little more 
closely, and is slightly more convenient for independent calcula¬ 
tion.* The value of 3.33 is almost universally adopted. The 
difference in the results by using these two coefficients is one-tenth 
of one per cent. 
The formula of Francis then becomes 
Q= 3.33 LH 3/ s 
or with the modification above mentioned, Q= 3 ^ LH 3/2 where the 
letters mean the same as above and with the same restrictions. 
The first one is the one properly termed the Francis formula. The 
second one is the one I have used, based on his work and agreeing 
slightly better with his results. 
Q represents the discharge in cubic feet per second. 
L and H are both measured in feet and decimals, not in 
inches. 
*The actual value from his results is 3 . 3318 . He adopted 3.33. 
3 1-3 is slightly more convenient for mental calculation and I have used 
it in the table in Bulletin 13 and since. 
