THE EQUILIBRIUM OF INCOMPRESSIBLE FLUIDS. 551 
being subject to the action of the forces X d m, Y dm, Z dm, and each 
element of the surface to that of the forces —p cos.Ads, — pcospds 
— pcoosyds, 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, —pds cosa, — pds cosp, — pds cosy, 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. 
Jf Xam —fpas cos A 
Svan—f pas COS 
Stam ~f pas COs ¥ 
which must vanish in case of equilibrium. This being supposed, we 
have 
J xan=f[pas cos A 
SXan=fpas eS en UP RM RAO (1°) 
Sftdma=f pas COs ¥ 
The integrals which contain the element (dm) of the mass are re- 
ferred to the entire volume of the liquid, and those which contain ds 
have reference only to the surface of that volume. The forces X dm, 
Ydm, Zdm, in consequence of being transferred to the origin of the 
coordinates, will give the couples (xv Y — yX) dm, (yZ—zY)dm, 
(xX — xZ) dm, which will be found respectively in the planes of 
xy,yz,zx. The forces —pds cosa, —pdscosp, — pds cosy 
will likewise give, in the coordinate planes, the couples 
— (xcos.A —ycos) pds, — (ycos.v —z cosm) pds, 
— (zcosA — x cosy) pds. 
The momenta of all the couples situated in the plane of ay being 
added together, and those of the couples situated in the planes of y z 
and z2 being likewise added together, all the couples will thus be 
reduced to three. 
S@¥-y) dm — f(x eos — y cos.) pds 
S (y%-2Y)dm — f (y cos .v— 2 00s) pds 
S X= #2) dn (zcos A — wcos.u) pds 
