THE EQUILIBRIUM OF INCOMPRESSIBLE FLUIDS. 551 



being subject to the action of the forces Xdm, Y dm, Zdm, and each 

 element of the surface to that of the forces — p cos . \ds, — p cos [uds 

 — p cos V ds, the equilibrium of the volume must be determined by the 

 mass of the invariable system. For this purpose we shall suppose that 

 the volume has become inflexible, and is invariably connected with the 

 origin of the coordinates : we shall transfer to that point all the forces 

 Xdm,Ydm,Zdm, —pdscosX, — p d s cos f/,, — p d s cos v, and 

 consider the couples to which this transfer will give rise. All the forces 

 transferred to the origin of the coordinates will be reduced to three. 



cosX 



COSf<, 



This being supposed, we 



J Xdm=- 1 pds cosX 



J Xdm = I pds cos/x (1.) 



/ Zdm = f pds cos v 



The integrals which contain the element (d m) of the mass are re- 

 ferred to the entire volume of the liquid, and those which contain d s 

 have reference only to the surface of that volume. The forces X d m, 

 Xdm, Z dm, in consequence of being transferred to the origin of the 

 coordinates, will give the couples (xY — yX) d m, (i/Z — zY) dm, 

 (x X — X Z) d m, which will be found respectively in the planes of 

 X y,yz,zx. The forces — pds cos \, —pds cos ij,, — pds cosy 

 will likewise give, in the coordinate planes, the couples 



— (arcos. A — y cos I/.) p d s, — (ycos.v — z cos f/.)pds, 

 — (z cos A — X cos v) p ds. 



The momenta of all the couples situated in the plane of xy being 

 added together, and those of the couples situated in the planes of y z 

 and z X being likewise added together, all the couples will thus be 

 reduced to three. 



/ (xY —yX) dm — I (x cos A — y cos .f/,)pds 



J {yZ — zY^ d m — I {y cos .V — z cos /x) p ds 



I {zX — xTj) d m — / (2 cos A — x cos .}x)pds 



