302 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
velocity V+W"; and besides expending in each second the quantity of energy RV in 
driving the ship ahead, it would expend the additional quantity RW" in driving the 
water astern. The relation between the sectional area and the velocity of the current 
produced by such a propeller is given by the following equation, 
R= "(V+W")W", (86) 
because g>B(V-f-W") is the mass of water acted on in each second, and W" the velocity 
impressed on it. The counter-efficiency , being the ratio in which the total work done 
exceeds the useful work, is 
W" 
1+ v (86 a) 
In previous writings* it has been shown that the amount of skin-resistance is pro- 
bably expressed by a formula of the following kind, 
( 87 ) 
in which dco is the area of an elementary portion of the skin of the ship, q— V(u 2 -j-v 2 -^-w 2 ) 
the ratio borne by the velocity with which the particles of water glide over that elemen- 
tary area, to the velocity of the ship (V), § the density of water, and k a coefficient of 
friction, whose value, as deduced from the performance of actual ships, is about '0036 
or ‘004 for a clean surface of painted iron. 
The integral §<fdu is called the augmented surface ; and the ratio 
is called the coefficient of augmentation. The denominator, )G dx, is what may be called 
the girth-integral, G denoting the immersed girth of a given cross section of the vessel. 
The augmented surface and coefficient of augmentation can be calculated for any parti- 
cular stream-line surface by drawing it, constructing such a diagram as that shown in 
tig. 5, and finding approximate values of the definite integrals by Simpson’s Rules; but 
to give exact general symbolic expressions for them involves difficulties which have not 
yet been overcome. 
The following are particular cases in which exact expressions have been found : — 
Indefinitely deep circular cylinder of radius l , ^=2 sin 0 ; 
Augmented surface per unit of depth, 2T§-£; 
Coefficient of augmentation, nearly. 
Sphere of radius l, q=\ sin 0; 
* Philosophical Transactions 1863, p. 134; 1864, p. 384; Civil Engineer and Architect’s Journal, October 
1861 ; Transactions of the Institution of Naval Architects for 1864, vol. v. p. 322; Shipbuilding, Theoretical 
and Practical. 
j) q 3 dco 
j G dx 
