PLANE WAVES OF SOUND 195 



The value of R will be sensibly the same as if the fluid were 

 incompressible. Its determination is therefore the same as in 

 the case of the steady flow of a liquid under pressure through 

 a capillary tube. In this case, if the section be circular, the 

 shearing stress per unit length on a coaxial cylindrical surface 

 of radius r is 27n* . ^"dujdr, and the resultant of the longitudinal 

 forces on the two curved faces of a cylindrical shell of thick- 

 ness 8r is therefore 



per unit length. The sectional area of the shell being Zirrdr, 

 the requisite pressure-gradient is 



, (31) 



drj 



which is independent of x. There being no radial motion, we 

 have dp/dr = 0, so that p, and therefore dp/dx, is also independent 

 of r. The equation (31) is then satisfied by u = A+Br !t t 

 provided B be properly determined. The constant A is fixed 

 by the consideration that there is no slipping at the wall 

 (r = a) of the tube. In this way we find 



The mean velocity over the area of the section is therefore 



?ra 2 .' o 9# &P>' ' 



Hence, for a circular section, 



12 = V/a (34) 



The formula (33) contains Poiseuille's* law of efflux of liquid 

 through a capillary tube, viz. that the discharge per second 

 varies as the pressure-gradient and as the fourth power of the 

 diameter. It may be made the basis of an experimental method 

 of determining p. 



* J. L. M. Poiseuille (1799 1869), a practising physician in Paris, who was 

 interested in the capillary circulation of the blood. The date of the memoir 

 referred to is 1844. 



132 



