440 MR. RENNIE ON THE FRICTION AND RESISTANCE OF FLUIDS. 
the resistance due to the asperities equal to R x U 2 , the sum of the resistance 
is R (U + U) 2 . 
M. Prony, applying his profound acquirements to the solution of all the 
cases of preceding authors, deduced from a selection of upwards of fifty experi- 
D Z 
ments the following simple formula : U = 26.79 V ; 
U being the mean velocity of the section of the pipe ; 
D the diameter of the pipe ; 
Z the altitude of the water ; 
X the length of the pipe : 
from which it appears that the velocity is directly in the compound ratio of 
the square roots of the diameter of the pipe and head of water, and inversely 
as the square roots of the length of the pipe ; that is, for any given head of 
water and diameter of pipe, the velocity is inversely as the square root of the 
length of the pipe. 
If we compare these results with those of Dubuat, Girard, and others, they 
approximate very nearly to each other. 
In general, if we incline a pipe to an angle of about 6^ degrees, or one ninth 
of its length, the discharge will be nearly equal to the discharge by additional 
tubes. The charge necessary to express the mean velocity of water issuing 
V 2 V s 
from straight pipes is by some authors equal to Dr. Young makes it 
the diminution of expenditure depending upon the contraction of the fluid 
vein and the friction of the pipe. 
The change occasioned by bends and angles in the direction of the fluid 
vein tends to diminish the velocity in a very remarkable manner. 
Dubuat undertook several experiments upon this subject, but the formula 
V 2 S 2 
proposed by him does not solve the difficulty, where gives the resistance 
due to one bend, V being the velocity, S the sine of incidence or reflection, 
and m a constant quantity determined by Dubuat to be 2998.50. 
Now although it is reasonable to suppose that the resistance should be pro- 
portionable to the squares of the sines of the angles of incidence, yet as all 
* Dubuat and Langsdorf. 
