STABILITY, PROPULSION, AND SEA-GOING QUALITIES OF SHIPS. 29 



Eeverting to tlie approximate formulae for small oscillations — 

 periodic time =2 — zzz ^2 t, suppose ; 



greatest angular velocity = — ^^ sin g 0, 



= — sm j 9, 



we see that the periodic time of the oscillation varies directly as the radius 

 of gyration, and inversely as the square root of the metacentric height. This 

 teaches us how to regulate the periodic time of a ship, either in settling her 

 design, or in the distribution of her weights. We see, for instance, that a 

 vessel with a rising floor and flaring sides tends to quick rolling, by having a 

 high metacentre ; that a cargo of railway-bars has the same efl^ect, by bring- 

 ing down the centre of gravity ; and that running-in the guns and sending 

 down the masts has a similar tendency, by decreasing the radius of gyration. 

 The expression for the greatest angular velocity has been sometimes inter- 

 preted as indicating that quick rollers roll through large angles. The fact 

 appears to be experimentally true, but its inference from this formula involves 

 reasoning in a circle. The formula only shows that for the same amplitude 

 the greatest angular velocity varies inversely as the time ; but this tells us 

 nothing about the amplitude, while the formula itself is obtained on the sup- 

 position that the amplitude is small. 



The position of the ship's centre of gravity and the length of the radius 

 of gyration cannot, practically, be obtained by calculation. The centre of 

 gravity is generally found by shifting some known weights through known 

 distances, and observing the angular motion. The displacement and meta- 

 centre are of course known by calculation, and the problem is then the same 

 as if the ship were suspended from her metacentre*. The radius of gyration 

 is found by observing the time of a small oscillation in stiU water, and 

 then eliminating the effect of resistance f. 



As the metacentre depends upon the moment of inertia of the plane of 

 flotation, it is different for pitching from what it is for rolling, and so for 

 any intermediate position :J:. Practically, the metacentre for rolling varies 

 from to 20 feet (as an extreme limit) above the water-line, while that for 

 pitching is from 70 to 400 or more feet high. The moment of inertia of the 

 ship also varies greatly with the direction of the axis about which it is 

 taken. 



Free HoVing in a Resisting Medium. 



The experiments of Messrs. Fincham and Rawson, undertaken at the sug- 

 gestion of Canon Moseley§, led to the conclusion that for vessels of semi- 

 circular section in which the disturbance of the water is the least possible, 



* The method, with an account of some experimental determinations on several of 

 H.M.'s ships, will be found in the Trans. Nav. Arch. vol. i. p. 39. See also v. p. 1 ; vi. 

 p. 1 ; vii. p. 205. 



t As to this, see Mr. Rankine's Note in Trans. I. N. A. vol. v. pp. .31, 32. 



\ See Dupin, ' Applications de Geometric.' He shows that the metacentric heights for 

 rolling and pitching are, in fact, only the two principal radii of curvature of the surface 

 of centres of buoyancy; and hence the metacentres for intermediate positions may be found 

 by the help of the ellipse of curvature. 



§ See Phil. Trans, for 1850, and Moseley's ' Engineering and Architecture,' pp. 616, 617. 



