296 Dr. T. H. Havelock. [Apr. 1, 
total resistance,* and the dotted curve is a parabolic curve inserted here to 
represent approximately the frictional resistance ; the difference between the 
two curves represents the residuary resistance, and is clearly of the same 
type as the theoretical curve in fig. 9. 
Fic.9 Fic.10 
Ca 
We can obtain a better estimate of equation (27) by taking an experi- 
mental curve for a model in deep water, and then building up curves for 
different depths. We must first put (27) into a form suitable for com- 
parison with deep water results. 
Limiting the problem to one of transverse waves only, the formula (27) 
must reduce to R = Ae~?*4, for h infinite and c = (speed in knots)/,/(length 
in feet). 
Writing v’ for v/,/(gh) we find ¢ = 11:3v’*h/L; thus although the actual 
critical velocity does not depend upon the length of the ship but only on the 
depth of water, the speed-length ratio (c) has a critical value which is 
proportional to the square root of the ratio (depth of water)/(length of ship). 
In (27) we cannot fix any value of v or ¢ and then calculate Rk directly ; 
we must work through the intermediate variable «xh. The equations may 
now be written as 
R= A (Kh)Pv'se-8 (1 —2eh/sinh 2h), (28) 
v2 =(tanhkh)/eh; 6’ =0218L/2; oc =1130'h/L. 
With h infinite this reduces to the previous form for deep water with the 
same constant A, so that a direct comparison is possible. As the velocity v 
increases from 0 to ,/(gh), « diminishes from o to 0; we select certain 
values of «xh, calculate the values from tables of hyperbolic functions, and 
thus obtain the set of values in Table VII, writing m for 
(«h)2v'4] (1—2ch/ sinh 2h). 
* J. Scott Russell, ‘ Edin, Phil. Trans.,’ vol. 14, p. 48, 1840. 
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