412 Mr Larmor, On Critical or [May 28. 



inertia increases or diminishes as the degree of heeling increases, 

 a criterion usually easy of application. 



6. In a very numerous class of cases we can completely 

 determine by geometry the curves mentioned above. Since any 

 surface of the second degree may be derived by successive ortho- 

 gonal projections, real or imaginary, from a sphere, it follows that 

 the locus of the centre of gravity of a constant volume cut off 

 from it by a plane is a similar and concentric surface. Hence for 

 all this class of surfaces, which includes quadrics, cones, cylinders, 

 pairs of planes, the surface of buoyancy is similar to the bounding 

 surface of the floating body. Now the radius of curvature of a 

 parabola or hyperbola is least at the vertex, and that of an ellipse 

 is least at the ends of the major axis, and greatest at the ends of 

 the minor axis : hence for paraboloidal and hyperboloidal surfaces, 

 and cones, and wedges, the critical position is really stable, and for 

 surfaces of which the section in the plane of displacement is elliptic 

 it is stable or unstable according as the major or minor axis is 

 vertical. 



Further, if the shape near the water line come under any of 

 these heads, the above conclusions clearly all apply, irrespective of 

 the shapes of the other parts of the body. 



7. We shall now investigate, as a typical case, the nature of 

 the oscillations of a body whose section at the vertex approximates 

 most closely to that of the parabola y 2 = ax, which rolls on a rough 

 plane, and in which G^isa very short distance c above 0. 



The equation of the e volute near the cusp is of the form 



2 16 „ 



J 27a ' 



and when it is displaced through a small angle 0, the distance of 

 the vertical tangent to the evolute from O is easily seen to be 

 equal to 



j<9 3 -csin6>, 



4 



to the third order in 6. 



Thus, if k represent the radius of gyration of the solid round 

 an axis through its vertex, the equation of motion is 



*" d 2 d t n B\ a m 

 ■jW = C [ e -6)-4 i e > 



to the third order in 9. 



And the flanking positions of stable equilibrium are given by 



6 = Jl2c/{Sa+2e). 



