384 
DR. W. J. MACQUORN RANKINE ON PLANE WATER-LINES. 
to be so small, compared with the dimensions of the body, as not to produce any appre- 
ciable error in the consequences of the supposition of motion in plane layers. 
(21.) Integral on which the Friction depends . — Suppose a portion of an oogenous 
neoid to be taken for the water-line of part of the side of a vessel, which part is of the 
depth cte, and that the resistance arising from friction between the water and the vessel 
is to be expressed — the law of that friction being, that it varies as the square of the 
velocity of gliding, and as the extent of rubbing surface. 
That resistance is to be found (as already explained in a paper on Waves, published 
in the Philosophical Transactions for 1863) by determining the work performed in a 
second in overcoming friction, and dividing by the speed of the vessel ; for thus is taken 
into account not only the direct resistance caused by the longitudinal component of the 
friction, but the resistance caused indirectly through the increase of pressure at the bow, 
and diminution of pressure at the stern, assuming the vertical disturbance to be unim- 
portant. 
Then for a part of the water-line which measures longitudinally dx, the extent of 
surface is 
u 
the friction on the unit of surface is 
KW(« 2 +® 2 ) 
*9 ' 
where W is the weight of a unit of volume of water, and K a coefficient of friction ; 
and that friction has to be overcome through the distance \/ u 2 +v 2 , while the vessel 
advances through the distance c, giving as a factor 
VvP + v 2 
Those three factors being multiplied together, and the result put under the sign of 
integration, give the following expression for the resistance, 
KWC \ 
R=— S 6Z 
2 9 
■sm~ 
dx. 
(46 a) 
Another form of expression for the same integral is obtained by putting - dy or f- d@ 
instead of c - dx ; and a third form by putting for the elementary area of the rubbing sur- 
face the following value, 
dz.— c - ■ dg; 
where dg is the distance between the asymptotes of a pair of orthogonal trajectories, as 
explained in article 19. This gives for the resistance 
t) KWc 2 
E =-ir fe 
( 47 ) 
