324 G. H. KNIBBS. ' 
assuming the zoll to be 261544 cm. Unfortunately the pressures 
were small and were determined by measuring the height in the 
reservoir of supply, instead of manometrically at two sections in 
the pipe itself ; so that although the experiments are numerous, 
their value for the purpose of very accurately ascertaining the 
form of the temperature function, and the laws of flow generally, 
is not high. The reduction of the head for the circumstances at 
the influx end of the tube is subject to some uncertainty as I have 
previously shewn from Poiseuille’s and Jacobson’s experiments,’ 
and as is generally known. Consequently the fall in pressure per 
unit length of tube—J or dP/dZ—is doubtful, and variable in 
amount, and the velocities opposite each temperature therefore 
require corrections in order to make them comparable. 
Hagen in discussing his results, treats independently the 
velocity law for the two main branches of his curves shewing the 
relation of hydraulic gradient to velocity. The first branch he 
recognises as belonging to the rectilinear régime and analyses it 
by the formula 
yd eo) etre ee ee (11 
s and ¢ depending on the temperatures of the water, and on the 
dimensions of the tubes. The term in U—the large term—is the 
Poiseuille’s law term : that in U7? is the loss of head at the influx 
end of the pipe.’ Passing over the intermediate régime, Hagen 
obtains by logarithmic methods, exactly as St. Venant did before 
him, the relation of the resistance head to the velocity in, and to 
the radius of, the pipes. Employing the formule 
h=kU™, whence log h=logk+n log U.........(12), (124) 
he finds for the three tubes the following values of n, viz., for 
A=177949 + ‘0690; for B 1-7393 + -0181; and for C 1:7987 + 
‘0168, taking the mean as 1-75. He further discusses the relation 
between m and the exponent of R and finally proposes for the 
turbulent régime the formula 
hakLRO™ (13) 
Stereos tewnee 
. 
1 Journ. Roy. Soc. of N.S.W., Vol. xx1x., pp. 98-103, 1805. 
2 See in place last cited, formule (3), (3a), (9), also pp. 102, 103. 
a 
