452 APPLIED MECHANICS 
width 6, of the weir, and the discharge over this portion is given by an — 
expression of the form k,b,h!. The width 6, will depend on the height h, 
and may be written 6, = mh. Hence for a ‘weir with » end contractions, 
where 7 is equal to 2, Ai or 0, 7 
Q = (hb, + nkyb,)hi = (kb — nkymh + nkymh)hi 
= {k,b — n(k, — k,)mh}hi = a(b — nBh)ai, 
where a and f are constants to be determined by experiment. 
The above formula Q =a(b—mfh)hi is known as Francis’s formula, 
and although it was first derived empirically from experiments, it 
will be seen from the foregoing that it has a rational basis. This 
formula is sometimes called the Lowell formula, from the fact that 
the experiments upon which it was founded were conducted at Lowell, 
in Massachusetts. The experiments of Francis were made on weirs — 
from 4 to 10 feet long, with heads varying from 0°6 to 1°6 feet, 
and the mean values of the constants « and £ were found to be 
3°33 and 0°1 respectively. Francis’s formula may therefore be written 
Q = 3°33(6 — 0-1mh)hi. F 
The head h in the discharge formule for weirs given in this Article — 
is usually taken as the head measured from the crest of the weir to- 
the still water level just above the weir, as shown in ‘ 
Fig. 743, and not as the depth over the crest. Generally 
the upper surface of the water drops and curves slightly 
before reaching the weir. In the experiments of Francis, 
the head was measured from the crest to the level of 
the water 6 feet above the weir. hol 
The effect of velocity of approach is considered in the next Article. 
391. Velocity of Approach in a Stream.—When the water in the 
stream has velocity before it reaches the weir, this velocity is equivalent — 
to an additional head at the weir. In order that 1 
the water in a stream may flow, its upper surface 
must slope downwards in the direction of motion, 
and the effective head at any point, neglecting 
friction, must be measured to the level of still 
water up-stream. Fig. 744 shows a longitudinal 
section of the stream in the neighbourhood of the Fic. 744. 
weir, and the horizontal line XX is the still water 
level up-stream. The height hy) is the additional head due to the 
velocity of approach. 
Reasoning as in the three preceding Articles, it follows that the 
ordinary formula Q = 2kb ,/2g-hi velocity of approach being neglected 
becomes Q = 2kb /2g{(h+h))'— 23} when velocity of approach is con: 
sidered. 
Also Francis’s formula becomes Q=3'33(b—0-1nh){(h + ho)? - ah 
when velocity of approach is considered. a 
Since hy is generally small compared with h+h , the term Aki is 0 ai ; 
neglected, and the ordinary formula then becomes Q = 3kb ,/2g(h + No)i 
and Francis’s formula becomes Q = 3°3(b — 0°1nh)(h + he). 
When it is not convenient to measure h, directly, the velocity of 
approach % may be computed approximately as follows. Let A denote 
