181, 182] METACENTER. 475 



Introducing the values of h x and hy we may write 



54) ' 



Since for a body of the shape of a ship 5T and # increase together, 

 we see that the larger x corresponds to the shorter time. The 

 pitching of a ship takes place more rapidly than the rolling. 



The locus of the center of huoyancy for all possible displacements 

 is called the surface of buoyancy, and the two metacenters are the 

 centers of curvature of its principal sections. Evidently the body 

 moves as if its surface of buoyancy rolled without friction on a 

 horizontal plane, for it would then be acted on by the same couple 

 as under the actual circumstances. 



182. Solid hollow Sphere and Cylinder under Pressure. 



We have dealt in 176 with a few cases of equilibrium of solid 

 bodies under stress, where the strain produced was homogeneous. 

 We shall now treat a number of cases in which the strain is not 

 homogeneous. If there are no bodily forces the equations 144), 175, 

 become 



55) 



Forming the divergence, by differentiating respectively by x, y, 2, 

 and adding, and interchanging the order of the operations d and A, 



56) (I + 2ii) 4<s = 0, A<5 = 0, 



so that <? is a harmonic function. Let us put (3 = a. If the strain 

 is pure, let qp be the strain potential, 169, so that 



rrr\ dtp dy d(D A 



57) = 35 t> = ^, = 87; *-^ = 



and (p is equal to the potential of a mass of density - > coinciding 



4 7t 



with the body under investigation. If this be a sphere or spherical 

 shell, we find, as in 125, 135, 



er>\ 



58) 



59) 9 ^^ + , 



where & is a second arbitrary constant, from which we have 



