180, 181] POSSIBLE ELLIPSOIDS. 471 



181. Equilibrium of Floating Body. Let us apply the equa- 

 tions of equilibrium to a solid body immersed in a fluid under the 

 action of any forces. Let us find the resultant force and moment 

 of the pressure exerted by the liquid on the surface of the body. 

 If we call the components of the resultant H 9 H, Z, and of the 

 moment L } M, N, we have 



S = / / p cos (nx) dS, 



42) H = I I p cos (ny) dS, 

 Z= I I pcoB(ne)dS, 



L = I I [yp cos (ns) zp cos (ny)} dS, 



43) M = / / {#p cos (nx) xp cos (nz)} dS, 



N = I I {xp cos (ny) yp cos (nx)} dS. 



If the body is in equilibrium it is evident that we may replace it by 

 the fluid which it displaces, which would then be in equilibrium 

 according to equations 4), and might then be solidified without disturb- 

 ing the equilibrium. 



If the body is only partly immersed we must apply the integration 

 to the volume bounded by the wet surface and a horizontal plane 

 forming a continuation of the free surface of the liquid and called 

 the plane of flotation. Over this plane p = 0, consequently the surface 

 integral is taken only over the wet surface, while the volume integral 

 is as before taken over the volume of the fluid displaced. With this 

 understanding we may couvert the surface integrals into volume 

 integrals taken throughout the space occupied by the displaced liquid, 

 that is, within the surface of the solid body below the plane of 

 flotation. We thus have 



441 



