632 
CONCLUSIONS APPLICABLE TO SHIP-BUILDING. 
26. If, when the work of any disturbing force begins to be done upon a floating 
body, it be already acted upon by some other force, which has caused it to incline 
from the position in which it would otherwise rest through some given angle a, as in 
the case of a squall or a heavy sea striking a ship already inclined by the action of a 
steady wind upon her beam ; representing by d the additional inclination given to it 
by the action of the disturbing force, whose work, in giving it this inclination, is re- 
presented by u{d), and representing the work done through this same angle 6 by the 
forces originally impressed, and still acting, upon it by U(^), we have 
Wc{vers — vers a)=\]{&)-\-u{&). 
Differentiating and transposing, 
U— {§U(^) l^esin (a + 6) ■ 
The small additional angle Id through which the body is made to roll by the appli- 
cation to it of a given small additional amount of work, 
varies therefore inversely as the sine of the inclination, and is greatest in the position 
nearest to the vertical position ; or, in other words, the body sustains in an inclined 
position a less change of that position by the application of a given disturbing force 
than it does in a vertical position. It yields most readily to the action of any dis- 
turbing force in its vertical position, and the further it is made to deviate from this 
position (within certain limits and under certain conditions), the more resolutely does 
it oppose any further deviation. This explains the liability of a ship to rolling when 
sailing before the wind, and her stiffness in the water when close hauled. 
Conclusions applicable to Ship-building. 
27 . To make an alteration in the angle through which a ship rolls, it is necessary 
to elevate or to depress her weights. In the former case she will roll through a 
greater, and in the latter through a less angle. It does not alter the amplitude of 
rolling to move the weights horizontally, but only the time of rolling, provided the 
trim of the ship remain unaltered ; for this does not alter the position of the centre 
of gravity of the ship or of the displaced fluid, and it is upon these that the stability 
of the ship depends (Art. 7 and equation 18.). 
28. When a ship’s motion is only a rolling motion, or about an axis parallel to her 
length, the position of that axis is, at any instant, determined by the intersection of a 
horizontal line through her centre of gravity, and a vertical line through the centre of 
gravity of her plane of flotation at that instant (Art. 19). 
29. A ship should be so constructed that the centre of gravity of that plane of 
flotation, whose boundary is the load water-line, may be vertically above the centre of 
gravity of the ship. If this be not the case, the pressure of the additional water dis- 
placed by any vertical oscillation of the ship, acting obviously at the centre of gravity 
