138 UNITEKSITY OF VIRGINIA PUBLICATIONS 



9. Before proceeding to the determination of the constants we return 

 for a moment to the expression (9) for the value of the resistance head. In 

 passing to the subsequent form of this expression we assumed AR to be 

 sufBcientl}' small to justifj' us in writing m/p equal to the mean hydraulic 

 radius: we propose now to esamiae the exact value of (9) expressed in 

 terms of Afi and the radius R of the pipe. We have 



o, = 7riR — AR)K 



Substituting in (7) we find 



[1+(1-P)^] jl-jp) V^ 

 ^^ = ^'' O^TTp 2g' (2*) 



where p = AR/U. Wlaence the exact value of the resistance head is 



Ai? [i+ji-py-] jl-jp) PL v- 



"- AL {1—pV " a 2g' ■ ^'^^' 



giving for m the coefficient of resistance, 



^^_AR [1+(1 — p)^] (1 — jp) 



AL (1—p)' 



which, if one prefers, may be written 



_AR 1 1 — (1— p)^ 



(36) 



(37) 



AL 2p (1 — p)* 



Tracing the curve, in rectangular coordinates, 



(l_.^)4y=|;l+(l_.^.)2] (i_|a;), 



we find y is positive between x ^ and x := 2, with a vertical asymptote 

 at a; = 1. The function y begins with the value 2 at a; = and increases 

 continuously from a; = to s ^ 1, at which latter value y becomes positive 

 and infinite, as x increases from 1 to 2, ?/ continuously decreases from 

 + 00 to 0. For all other positive values of x the values of y are negative. 



In the expression for m previous to this article the value of m continually 

 increases as r or V or both decrease indefinitely, and becomes infinite when 

 either of these variables vanish. If AiJ were the thickness of a solid obstruc- 

 tion it is obvious that the superior limit of AR would be R, at which the 

 pipe would be completely closed and necessarily v would vanish. But AR 

 is as we know merely a form for an equivalent thickness of a liquid 

 disturbance represented by a contraction of stream lines, possibly represent- 

 ing an oscillatory tranverse vibration of fluid particles. Moreover the form 



