132 UNIVEESITY OF VIRGINIA PUBLICATIONS 





This is Chezy's fonnula for the resistance head, in which 



„ dE 



dL 



represents his coeiBcient of resistance and which he assumed to be a 

 constant. But, the results of experiment show that we have fallen into 

 error in passing to the limit in the process of integrating as for a con- 

 tinuum. This perfect condition of affairs does not exist in nature and 

 the result obtained above is only a first or roughly approximate statement 

 of the. facts. "VVe proceed therefore more cautiously to the finite summation 

 of the finite differences. 



4. Let there be n uniformly distributed annular rings of easy curvature, 

 of equal apertures and area <o, and distant apart AL. Let v be the mean 

 velocity parallel to the pipe axis of the water passing through o, and V 

 that passing through the unobstructed area O. The loss of head at a 

 single contraction and adjacent expansion is 





C) 



If AR represents the thickness of the ring and ^ is a mean perimeter 

 such that p AR = w, then the above loss is represented, after putting 



n = 0) + p. AR, 

 by 



A/i = ( 2AR-4-^-AR- \^~: (8) 



The loss of head due to the « obstructions, where n . AL = L, is 



pL V 



'=-'-(-:^^-^.^) 



^- <»> 



If we assume that the thickness of the interference AR is small, which 

 the experimental results show to be the case for smooth pipes, we may 

 write without serious error P/n for p/w and for either of these the 

 reciprocal of the mean hydraulic radius r. Also putting 



2—= /,;, (10) 



AL 



