288-289] PROBLEMS OF STEADY MOTION. 521 



The equation (1) might have been obtained from Art. 286 

 (4) by direct transformation of coordinates, putting 



The integral of (1) is 



logr + B ............... (2). 



Since the velocity must be finite at the axis, we must have A = 0; 

 and if we determine B on the hypothesis that there is no slipping 

 at the wall of the pipe (r = a, say), we obtain 



x-V-*") ..................... (3). 



This gives, for the flux across any section, 



^.P^.. ...(4). 



Jo 



It has been assumed, for shortness, that the flow takes place 

 under pressure only. If we have an extraneous force X acting 

 parallel to the length of the pipe, the flux will be 



In practice, X is the component of gravity in the direction of the 

 length. 



The formula (4) contains exactly the laws found experimentally 

 by Poiseuille* in his researches on the flow of water through 

 capillary tubes ; viz. that the time of efflux of a given volume of 

 water is directly as the length of the tube, inversely as the 

 difference of pressure at the two ends, and inversely as the fourth 

 power of the diameter. 



This last result is of great importance as furnishing a conclusive proof that 

 there is in these experiments no appreciable slipping of the fluid in contact 

 with the wall. If we were to assume a slipping-coefficient /3, as explained in 

 Art. 285, the surface-condition would be 



or w= \dwjdr 



* "Recherches experimentales sur le mouvement des liquides dans les tubes de 

 tres petits diametres," Comptes Eendus, tt. xi., xii. (1840-1), Mem. des Sav. 

 Etrdngers, t. ix. (1846). 



