ON THE FLOW OF WATER IN PIPES, CONDUITS, ETC. 137 



where 



/VT V" 



+ brP 



tan® 



Jr 



4a 



(27) 



1 



64aJr2 



the values of ® lie between and tt. 

 Whence 



c = -^=, VI + brfi coti©, (28) 



8 



y = ^coti® V (1 + ir^)Jr. (29) 



VI' 



8. In eliminating m between (20) and (24) there results 



y = ^9JL- l±bl^. (30) 



A l — a+-^Vr 



A 



Hence for capillai7 tubes, in which &r/3 and— Vr are negligible compared 



A 



with 1, we have 



^ 2gJ_ ^., (3j) 



A(l — a) ' ^ ^ 



or the velocity of discharge varies as the square of the diameter and whence 

 the quantity of flow as the fourth power of the diameter. TMs result was 

 established by Poiseuille in a series of experiments conducted with the 

 most elaborate care and precision; see Annales de Chemie, III, xxi, 76. 



It appears from the above that in tubes of extremely small diameter 

 the velocity of flow is independent of the coefficient of rougixness with 

 respect to length, which is the most important factor for pipes of or dinar)- 

 diameters. The value of m for r very small is 



(32) 



1 + aV rV 

 Hence the head lost by pipe resistance is 



'' = 1 + aV l^g ' ^^^) 



which indicates that the ratio of the lost head to velocity is nearly constant 

 for any given tube. A result verified by the experiments of Beynolds, 

 Philosophical Transaciions (1883, Part III.). 



