134 UNIVERSITY OF VIRGINIA PUBLICATIONS 



the slope of resistance with respect to the length of the pipe, aiid 



the coefBcient of contraction with respect to the cross-section. That AR/'AL 

 and Ai? are not constants for a given pipe or channel is quite evident if 

 our theory be sound.' It would appear reasonable that the more swiftly 

 the water is flowing with mean motion through the pipe the flatter would 

 be the disturbed lines of flow and the smaller would be the thickness of the 

 disturbing fluid obstruction, which fact is completely established by experi- 

 ment in all ordinary' cases of regular boundaries. It appears also from 

 experiment that 



AR 



AL 



not only decreases as the velocity increases but also as the mean radius 

 increases. In other words it is a decreasing function of the two variables 

 V and r, and is of the form 



— • (15) 



f(y,r) 



where f is a positive and increasing function of V and r. Moreover as it 

 appears incredible that the mean slope of the roughness^ of disturbance 

 AR/'aL should become infinite in any ordinary ease when Y and r become 

 arbitrarily small, we conclude that the function f{V,r) has a positive 

 inferior limit /(O, 0). \Viiile the form of the function / is certain to be 

 verj^ complex in its complete expression, we feel justified in writing it 

 equal to the first three terms of its expansion in positive powers of the 

 variables V and r. Thus 



f{Y, r) = f{0, 0) + AVo- -j-Br^, (16) 



wherein A and B are constants and a and ^ are jDositive numerical para- 

 meters but slightly dependent on V or r. 



On substituting (16) in (15), then dividing the numerator and denomi- 

 nator by the positive constant f ( 0, ) , there results 



1 + (z7« -f br^ ' 

 wherein 



_ 2 _ A B 



>"- f{o,oy ''~f(o,oy "- f(o,o) 



