110 G. H. KNIBBS. 
expression for the total flux q across the section and integrating: — 
thus,— 
R nF it” 
— 9, oe 
q = 2r 2h urdr SL 
‘The expression for the mean velocity U in the section is found by | 
dividing that for the total flux by the sectional area S=7?, : 
thus ; ; 
For this régime therefore the resistance to flow varies as 1/ R? 
11. Flow in an elliptical cylinder.—Capillary tubes have usually { 
been assumed to be right circular cylinders, and their radii have 
been determined on that assumption by filling definite lengths 
with mercury, ascertaining its weight, and from the weights the 
volumes of the cylinders. Poiseuille however did not rely 00 
this method but made direct measurements. Discs of about 1 
millimetre in thickness cut from the tubes, and prepared ee 
grinding and polishing, were mounted with Canada balsam : 
between plates of glass.2 Under the microscope the contour # : 
the cavity of the tube appeared asa slightly opaque line, permitting 
the measurement of diameters. In thirty-four sections thus pre 
pared only one instance is recorded of uniform diameter, ea 4 
greatest difference of diameters being an instance in which ie 
ratio was 1} to1—Tube F. On the supposition that the sect? 
were ellipses Poiseuille used the mean proportional of the diameter 
—t.e., the diameter of a circle of equal area—as the equivalent 
mean. As however the mean velocity varies as the square of the 
diameter, see (16), this procedure, it is easy to see, will be inexact 
when the ellipse differs sensibly froma circle. We proceed theme 
fore to determine the equation of the flow in an elliptical cy linder 
1 This development is that of ordinary treatises or hy drodyn® 
cleared of the irrelevancy as to a boundary condition apparently wee 
realized, at any rate with glass tubes: qis of course the volume of the fo¥ 
across a section, in a unit of time. 
2 Mém. des Savants étrangers, t. 9, p- 456 et 457. 
