60 Lord Rayleigh on the Question of 



we have the three relations 



— 2 = x + 2y — 3z + n, 



— 2=— ?/ — n, 



l = z, 

 so thnt 



,7' = n — 3, i/ = 2 — n, s=l, 

 and 



Pa^ 3 p.(4)" (1) 



Since ?i is here indeterminate, all we can infer from dyna- 

 mical similarity is that 



P=^ C -V/(W"), • ( 2 ) 



where /is an arbitrary function. 



For capillary tubes and moderate velocities P varies as the 

 first power of iv, so that in (1) n==l. In this case 



F = Avc-*pw, (3) 



A being an arbitrary constant. When, on the other hand, 

 cw/v is great, experiment shows that n=2 nearly. If this 

 law be exact, (1) gives 



P = Bc- 1 /M0 2 , ( 4 ) 



independent of v. The second power of the velocity and 

 independence of viscosity are thus inseparably connected. 



In the above theory no account is taken of any variation in 

 the walls of the tubes. Either they must be perfectly smooth, 

 or else the irregularities must be in proportion to the diameters. 

 Under this limitation (2) would appear to hold good, at least 

 if there be no finite slip at the walls. 



The proportionality to p, expressed in (4), has probably not 

 been tested experimentally. Neither is there any complete 

 theoretical deduction of (4). But a comparison with Torri- 

 celli's law of efflux is significant. The resistance is the same 

 as if it were necessary to renew continually the velocity of the 

 liquid at intervals which are proportional to the diameters of 

 the pipes. 



The connexion between the alteration in the law of resist- 

 ance and the transition from regularly stratified to eddying 

 motion has been successfully traced by Reynolds. The ques- 

 tion is, Why do eddies arise and take possession ? From the 

 description and drawings given by Reynolds it is natural to 

 suppose that in the absence of viscosity the stratified motion 

 would be unstable, and that it is stable in small tubes and at 

 low velocities only in consequence of the steadying effect of 

 viscosity then acting at an advantage. It was with this idea 



