30 REPORT — 1869. 



the dynamical stability found by experiment differed very little from that 

 derived from the rise and faU of the centre of gravity ; but in the case of 

 a model of txiaugular section, the stability found by experiment was in 

 defect. In the semicircular model the extreme inclination produced by the 

 sudden application of the force was, with a fair degree of approximation, 

 double that due to its statical effect. "With the triangular model the extreme 

 was less than double the statical inclination. This is nothing more than 

 might be expected from the disturbance of tlie water which would be set up 

 by the angular model, and which would, of course, take up part of the work. 

 But this experimental confirmation of theory is highly satisfactory ; and, 

 however we may now look back upon the matter, it is really upon these ex- 

 periments that the confirmation of oiir theories rests. 



In a resisting medium, the amplitude of the oscillations is very quickly 

 affected, but the periodic time undergoes very slight change. But the period 

 is altered to a slight extent. On this subject we refer, first, to the account 

 given by Poisson, Stokes, and other writers on mechanics, concerning the 

 oscillation of a pendulum in air ; secondlj', to Mr. Troude's experiments * on 

 a pendulum oscillating in water ; and thircUy, to Professor Eankine's paper 

 on keel-resistance t) ia which the measure of diminution is given on a certain 

 hypothesis. 



Bessel and Poisson have pointed out that the virtual loss of weight due to 

 oscillation in a resisting medium is greater than that due to the mere im- 

 mersion. Mr. Moseley makes the same remark with reference to the rolling 

 of ships. 



^ Professor Bankine has investigated the effect of the steadying-action of a 

 keel on the rolling in smooth water, on the assumption tliat the moment of 

 the righting couple is simply proportional to the inclination, and also that 

 the moment of resistance to rolling, caused by the action of the water on the 

 keel and floor, is proportional to the angular velocity. He finds t that the 

 periodic time is altered from 



2 TT K 2 TT K 



to 



V 



9F 



"\/^-Sp 



where c is a constant depending on the moment of the resistance ; so that 



moment of resistance of water 

 displacement X angular velocity' 



the effect of the resistance thus lengthening the periodic time in the same 



* Trans. Inst. Nav. Arch. vol. iii. p. 31. Mr. Froude bas there shown that when a 

 pendulum or ship performs isochronous oscillations in a medium the resistance of which 

 varies as the square of the velocity, the amplitudes of the successive oscillations, as reduced 

 by resistance, will form successive ordinates of a curve, which approaches, with a great 

 degree of exactness, to an equilateral hyperbola, referred to one of its asymptotes, equal 

 periods of the oscillations being represented by successive equal increments of the abscissa. 



The experiments with a pendulum as exhibited in the diagram (plate 2 of the volume 

 referred to) accord very closely with the law which may be thus expressed :■ — If 6„ be the 

 initial amplitude, and 6^ that of the mth oscillation, then that at the end of any other, say 

 wth will be 



n W 0„ 9m 



{m — n)i)m + n ^o 

 t Trans. Inst. Nav. Arch. vol. v. pp. 30, 31. % Ibid. 



