1883.] 



"Apparently Neutral" Equilibrium. 



411 



really stable. But there is this peculiarity, that its oscillations 

 round the vertical are no longer approximately (Simple Harmonic, 

 but follow another law which we can easily investigate, and that 

 they are executed with extreme slowness : and we can trace the 

 change to this new law from the rocking motion which is com- 

 pounded of oscillations round the two flanking stable positions 

 alternately. 



3. If the cusp pointed upwards, - . - were a very short 

 distance below 0, we would have a 



vertical stable position flanked by 

 two very near positions of instability : 

 and so, when G moves up to 0, the 

 vertical position becomes unstable. 

 It is important, then, to bear in mind, 

 that cases which satisfy the condition 

 of stability, but are near to the criti- 

 cal case, are practically unstable for 

 oscillations of any considerable a- 

 mount, when the radius of curvature is a maximum at A. 



4. These considerations clearly apply to all cases of critical 

 or "apparently neutral" equilibrium, so that the determination 

 of its real character carries with it the determination of the 

 practical character of all other cases which approximate to that 

 condition. 



5. In the case of a floating body this discrimination is easy. 

 If we consider, as usual, oscillations in which the displacement is- 

 constant, the centre of gravity of the displacement traces out a 

 surface called the Surface of Buoyancy, and we know that the 

 tangent plane to this surface corresponding to any position of 

 the oscillating body is always horizontal, and that therefore the 

 resultant fluid pressure acts along the normal at its point of 

 contact. The circumstances of the oscillation are therefore the 

 same as those of a solid body with the same centre of gravity, 

 but bounded by the surface of buoyancy, and rolling on a friction- 

 less horizontal plane. The evolute of the section of the surface of 

 buoyancy is the locus of the metacentre, and has been called 

 the Curve of Stability : and a case of equilibrium which ap- 

 proximates to the critical condition will come under § 2 or § 3 

 according as this curve has its cusp pointing downwards or up- 

 wards. Now the radius of curvature of the surface of buoyancy 

 is known by the ordinary theory to be equal to the moment of 

 inertia of the corresponding plane of floatation about an axis 

 through its centre of gravity perpendicular to the plane of dis- 

 placement, divided by the volume of the displacement: and there- 

 fore the case comes under § 2 or §3 according as this moment of 



VOL. IV. PT. VI. 



29 



