ON THE FLOW OF WATER IN PIPES, CONDUITS, ETC. 131 



3. In the case which we surmise we suppose an indefinite number of 

 equal annular contractions uniformly distributed along the length of 

 the pipe, and that these effectively represent the equivalent of the choking 

 of the pipe by eddies, vortices, regular or irregular as the case may be, 

 whose effect is to produce a loss of energy and corresponding retardation 

 in the flow. But now the volume of water moving in a contracted section 

 is commensurable with that moving in the adjacent uncontracted portion, 

 and hence the loss of energy caused by a single one of these contractions 

 and its adjacent exjjansion is not to be measured as an impact but directly 

 by the change or difference of the kinetic energy of these two masses. 



If we fall into the error of assuming the contractions to be infinite in 

 number and close together and each contraction to be infinitesimal, we 

 would proceed as follows. The element of the loss of head at a cross- 

 section of area w at a distance I from the entrance of the pipe is equal to 

 the differential of the kinetic energy, or 



dh=^^. (1) 



But V being the velocity of discharge and O the actual cross-section of 

 the pipe, the law of "continuity" requires 



vo> = Fn = constant. 



dv = —~do,. (3) 



Let dr be the mean thickness of the obstruction and p a perimeter such 

 that p dr ^ w. Then 



dh= — ~^dr. (3) 



If L is the total length of the pipe, the loss of head due to the resistances 

 inside the pipe is 



''^- v^ p dr 



_n= f "^J^^di. (4) 



J q w di 

 If we represent the mean values, throughout the pipe length, of v, p, ta. 



dR 



the loss of 



dR P V- ^ ^ 



-ryby y, P, ^,~jf the loss of head, dropping the negative sign, is 

 di d-L/ 



h 



dL a r/ f dl. (5) 



