ON THE FLOW OF WATER IN PIPES, CONDUITS, ETC. 139 



in which it occurs in m is as a mean value of such a thickness or range of 

 disturbance for the entire pipe length. Would it be possible therefore, 

 under any real conditions, for finite values of R and H, for A.B to be equal 

 to R and thus m = oo and 7 = 0? Such a condition if even approximately 

 realized could only be attained under small values of R. Furthermore, 

 could it be possible for AR to exceed R in value and have as its superior 

 limit 2R or D the diameter of the pipe? If so, then for values of the ratio 

 ARy/R increasing from 1 to 2 the values of m would diminish, becoming 

 ultimately zero. In this speculation as to the possible behavior of AR 

 we have not considered the behavior of the co-factor AR/AL in (36) ; 

 that .for capillary tubes this does not becomes arbitrarily great is assured 

 by the experiments of Poiseuille. It is, moreover, possible that under these 

 circumstances it may become arbitrarily small when its co-factor becomes 

 gi-eat and thus the value of m attain a finite superior limit between and 2, 

 in the neighborhood of AR = B. It is altogether unlikely that, however 

 AR/R may behave, AR/R can ever in any real case of flow actually attain 

 the value 1 and m thus the value oo. Some writers on hydraulics point 

 out as a fact that under certain circumstances of relationship of r and s 

 the coefficient m decreases with decrease of r and s; see the remarks on this 

 matter in Hering and Trautwine's translation of Kutter's Flow of Water, 

 with reference to the flow in open channels. The writer has not been 

 able to get access to any reliable experiments on tubes or pipes which 

 exhibit this contrary result in any manner suggestive of law, with the 

 exception of a group of four experiments by Eennie on glass tubes of 

 respective diameters .002; .004; .006; .0083 feet, quoted by J. T. 

 Fanning in his Water Supply Engineering, p. 239. Corresponding to a 

 velocity of about 10 feet per second. Fanning gives as the corresponding 

 values of m for these four pipes respectively about 



.00060; .0020; .0043; .0043, 



which are entirely at variance with any formula for m or any experiments 

 known to the writer on pipes. 



Granting it to be a fact that for certain values of B. and R the values of 

 m decrease as R decreases, could the explanation be that the representative 

 equivalence of the range of the disturbance AR becomes greater than R 

 and that under such circumstances there is a reflex interference of the 

 disturbances with each other or transverse vibrations of the liquid particles 

 with the effect that their reducing components of velocity parallel to the pipe 

 axis annul each other and thus cause a relative augmentation of the flow 



