112 G. H. KNIBBS. 
The integral for the total flux across the section is therefore 
B 
tn he uy dy 
in which w must be written as in (17) omitting the term nt 
Making this substitution, writing P/(7Z) for A, and integrating 
we obtain 
3 3 
ce a (18) 
4 LB +C? 
so that if we put 
1 
atee.2:)' (19 
Re = (55) (19) 
R, will be the radius of a circular cylindrical pipe of equal efflux. 
From (18), since S=~BC we obtain for the mean velocity in tit 
section 
Weide te PO... 
Mae RE: ae ae Be Fe | 
The radius R’, of a circular cylindrical pipe of equal mean velocity : 
is therefore 
...(20) 
a BO: (21) 
1 {3(B? + 07)} | p | 
When B= € these formule become identical with those for right | 
circular cylinders, viz. (2) and (16). Poiseuille tacitly assum® 
that R, = R’, = v(B C). The error of this is best seen bY 
reéxpressing (18) in the form of a series,! and comparing “ 
result with B?C? similarly developed. Putting i 
R, = $(B+C) ande = Boe 
B+C 
(18) may be recast in the form 
.wPR; a 
= e 1 Hee 2 ee Bat Ne ee 18a 
q Ba (1 = 4e° + Te* — 8e* +...) 20... ( ) 
If the radius of the circle of equal area be used, we have 
BC* = Bt (1 - 2c? +€*) ae 
The difference in relation to the degree of precision to whieh 
Poiseuille has expressed his results is sonietimes appreciable. al 
example in the case of the tube F, assuming the differene® 
See ne ee rere epee 
1 This gives also a Practically useful formula for computations. 
