328 : G. H. KNIBBS. 
9. Defects of the ordinary velocity formule.—Brahm’s and Chezy 
are said to be the authors of the usual formula for flow in pipes 
or channels, viz., 
i de eS ee 
an expression which seems to have served as a mould for the great 
majority of evaluations of velocity, and which is generally known 
as the “ Chezy formula.” 
Darcy! and Bazin’s’ modification of this, in view of an obvious 
defect in a formula proposed by de Prony,’ was 
U = J = WIE LY sic sge ios: (15) 
a+ Bi 
R 
a and B being coéfficients which varied with the roughness of the 
surface of the pipe or channel, a factor wholly ignored by de ‘Prony. 
Both Darcy and Bazin recognised that this expression did not 
represent the phenomena in all their generality, but regarded it as 
a sufficient approximation for practical purposes. 
The limitations of this formula were studied by Ganguillet and 
Kutter,* who, in order to embrace all sizes of pipe or channel, 
developed, in a most ingenious manner, the empirical expression 
be 
eG a+ y =z : : 
1+(a@+ 5) a 
in which a, 6 and care constant for every case, and y is a coéfficient, 
depending upon and increasing with the roughness of the boundary, 
and R is the hydraulic radius = } R in the case of a pipe. For 
C.G.S. units a= 230, 6=10, e=-0155 and y varies between 006 
VERE EY ccc es (16) 
'¥ Rooheeohes expérimentales relatives au movement de ’eau dans les 
tuyaux.—Mém. des Sav. étrang., t. 15, pp. 141 — 408, 1858. 
2 Recherches expérimentales sur l’écoulement de l’eau dans les canaux, 
découverts.—Mém. des Sav. Etrang., t. 19, pp. 1-494, 1865. 
s rf oo physico-mathématiques 1790. The formula was aU + 
# A general formula for the uniform flow of water in rivers and other 
channels. ‘Trans. by Hering and Trautwine. Macmillan, 1889 
