80 A THEORY OF FLUID FRICTION. 
The resistance R is obtained when the length is known:— 
: 
R=ff)V?. dl=F.tan ag.L=w.b.h 6) 
Table 1, Plate 56, giving the functions for determining frictional resistance, 
was obtained by assuming a value of 1.00 for tan a at velocity Vo = 1.00. The 
values of J, the length between the intervals of velocity; of L, the total distance 
aft of the cutwater; of b, the breadth of the wake; of rz, the resistance due to 
friction corresponding to the interval in length /; and of R, the resistance corre- 
sponding to the total length L, can be found from these functions as follows :— 
; Ve 
h = function 2 x —* (7) 
25 
6 = function 4 x Yo (8) 
tan @ 
3 Ve 
hx 6b = function hx function 6 x ——*— (9) 
ag.tan a 
mean tan a = function mean tan a x tan a (10) 
hy 1 3 3 
LS Ua fiction La x SG = = function / SLU (11) 
mean tana function mean tan a” 22. tan’a ag .tan’a 
EXD : wv, 
[ay SU AR CAEN 
f/ mean tan a function? x 22. fF. tan’a (x2) 
: we? 
Z = function Z x Fe P ae (13) 
3 
vy = function / x function mean tan a x sneiGal 
2g. tan a 
3 
= function (% x 4) x Tyo (14) 
ag. tan a 
R= rr (15) 
For the same roughness of surface, remembering that tan a varies as Vo’, we 
find from the preceding equations that— 
b varies inversely as Vo. 
h varies as Vo. 
L varies inversely as Vo. 
R varies as Vo. 
