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PROFESSOR W. J. MACQUORN RANKINE ON THE 
increased submergence of the body ; but the speed, and consequently the angle of diver- 
gence, of the waves of the first class is unaltered, because they depend on the time occu- 
pied by the solid in moving through a certain portion of its length. 
§ 19. Remarks on the Skin-resistance. — It is well known through observation: — that 
the friction between a ship and the water acts by producing a great number of very 
small eddies in a thin layer of water close to the skin of the vessel, and also an advancing 
motion in that layer of water ; that this frictional layer (as it may be called) is of insen- 
sible thickness at the cutwater, and gradually increases in thickness towards the stern, 
by the communication of the combined whirling and progressive motion to successive 
streams of particles ; and that, finally, the various elementary streams of which the fric- 
tional layer is composed, uniting at the stern of the ship, form her wake — that is, a steady 
or nearly steady current, full of small eddies, which follows the ship, but at a speed 
relatively to still water which is less than the speed of the ship. 
The central stream of the wake has the greatest velocity ahead ; and other parts of it 
have velocities diminishing from the centre towards the circumference. If the friction 
between the water and a given area of the skin of the ship is equal to that of an equal 
area of one layer of water upon another at a given velocity, the mean forward velocity 
of the whole wake, relatively to still water, and its mean backward velocity relatively to 
the ship, are each of them equal to one half of her speed. 
The effect of discontinuity of form, as when the figure of the vessel presents angles to 
the water, is to produce eddies which are dragged along with the ship, and thus to add 
to the wake ; and hence the resistance arising from discontinuity of form is analogous 
in its laws to that arising from friction ; and both those forces are comprehended under 
the name of eddy-resistance. Bodies of discontinuous forms, however, are foreign to the 
subject of this paper. 
Let V, as before, be the velocity of the ship; let W' denote the mean velocity of the 
wake, and C its area of cross section, both taken at a distance astern of the vessel suffi- 
cient for the wake to have become a steady forward current. Let R be the amount of 
the skin-resistance in units of weight, and the density of water. Then the mass of 
water added to the wake in each second is g>C(V — W'); and the velocity impressed on 
that mass by the force R is W' ; whence we have the following equation, 
R=I. f C(VW'-W' 2 ); (84) 
and if the mean velocity of the wake is half the velocity of the ship, that equation becomes 
(84 a) 
It is obvious from equation (84) that, for a given amount of skin-resistance, the wake 
has the least possible sectional area when its mean speed is half that of the ship. 
The work done by the ship on the water per second in producing the wake is RV ; 
the actual energy of the current of the wake is increased in each second by the amount 
