STABILITY^ PROPULSION, AND SEA-GOING QUALITIES OF SHIPS. 27 



the plane of symmetry, or middle-line plane, as it is teclinically called. The 

 limiting position of this intersection, when the angular deviation is indefi- 

 nitely small, is called the metacentee. This metacentre is the critical 

 point below which, if the centre of weight be kept, there will be stable 

 equilibrium. 



It is shown in books on hydrostatics that if a floating body receive a small 

 inclination, the two water-sections intersect in a line passing through the 

 centre of gravity of each, and also that the Kne passing through two suc- 

 cessive centres of buoyancy tends to parallelism with the water-section. It 

 follows that the stability of a ship, statically considered, may be measured 

 by the statical stability of a solid, whose centre of gravity coincides with 

 that of the ship, but whose surface, instead of floating in water, rests on a 

 horizontal table. This representative surface is the surface formed by the 

 centres of buoyancy of the ship at different inclinations. The metacentre 

 of the ship is then the centre of greatest or of least curvature of this repre- 

 sentative surface, called the surface of buoyancy, according to whether we 

 consider transverse rolling or longitudinal pitching. 



When we pass from statics to dynamics, the righting or upsetting force 

 simply represents an acceleration. But if the ship be considered as concen- 

 trated at its centre of gravity (in disregard of the actual distribution of 

 weights in respect of inertia), the same geometrical considerations hold, and 

 the space through which the centre of gravity rises or falls as the surface of 

 buoyancy rolls is called the measure of dynamical stability*. It is simply 

 proportional to the integral of the statical stability taken with reference to 

 the angle of inclination. Its product into the displacement gives the mecha- 

 nical ivorl- required to heel the ship, considered as concentrated at its centre 

 of gravity, to a given angle. An example of its use is in the solution of the 

 problem of finding how much a ship would lie over to a sudden gust, strong 

 enough, if it came on gradually, to heel the ship to a given angle. The 

 rough solution is that she woiild lie over to double the angle of the statical 

 stability ; and this remark is of importance in judging of the safe limits of a 

 ship's stability. This solution, it is to be observed, takes no account of the 

 moment of inertia of the ship about its centre of gravity, and very little 

 account of external form. 



Experiment and theory both go to prove that the time in which a ship per- 

 forms a complete double oscillation varies but very little, whether the am- 

 plitude of the oscillation be small or large. Hence every ship has its equi- 

 valent pendulum. If k be the radius of gyration of the ship, /x the distance 

 between the metacentre and the centre of gravity, the length of the equiva- 



lent pendulum is — , the periodic timet is 2 , and the greatest angular 



^ Vi'/^ 



velocity is — zJUli sin | 6, where d is the amplitude, or departure from the 



vertical ; but the approximation in this last formula is much less than in that 

 for the time. 



Dupiu has shown that the free roUing of a ship, regarded without refer- 



* The true dynamical stability is the actual work done in heeling ; but the words are 

 ordinarily used in the sense stated in the text. 



+ The time here used is that of a double oscillation ; i. e. the time which elapses be- 

 tween the bob of the pendulum passing the lowest point twice in the same direction. 

 There is very often confusion between double and single oscillations, both with analysts 

 and in the records of experiments. 



