502 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



coordinates have particular values, it follows that (p{x,y,z) will contain no terms 

 such as Ao?, B«/, C;s ; and consequently that its differential coefficients, viz. 



d (f) d <p d <p . 

 d s' dy^ dz' 



will vanish at the centre of gravity. Wherefore, in all problems of this class, the 

 foregoing theorem may be applied to the equation of the surface of the fluid, since 

 the necessary conditions are fulfilled. 



Now attending solely to the equation of the surface, viz. 



it has been shown that the expressions 



d <p d <^ d <f> 

 d x^ dy^ dz^ 



represent forces respectively parallel to the coordinates, the resultant of which is di- 

 rected perpendicularly towards the surface. If it be supposed that every particle of 

 the fluid is urged by forces expressed by substituting its coordinates instead of the 



coordinates of the surface in the same functions j^, ^, j^, it is proved in the theo- 

 rem that the mass will be in equilibrium, and may be divided by an infinite number 

 of level surfaces into thin strata that exert a constant pressure upon one another. 

 We have, therefore, now to inquire how the equilibrium which takes place when 



j^, 7^' Tz ^^^ *^^ forces in action, is to be preserved when, instead of these, the 



d ($>' d (f)' d <p' 

 Other forces, -^., j-, -j^, are substituted. These latter forces may be considered as 



produced by additions made to the first, and they may be thus written, 



d <^ /d<p' ^ d_^\ d_^ /d^ ^\ d_^ \ {^^' ^ ^\ 

 d x~^ \dx (I x)^ dy ~t" \dy ~ dy)' d z "^ ydJ ~ Tz) ' 



and supposing the whole body of fluid to be divided, as in the theorem, into thin 

 level strata, to which the joint action of the forces -^, -j^, -j- is at every point per- 

 pendicular, it is evident that the equilibrium will be destroyed when the additional 

 forces come into action, unless their resultant, urging any particle, be perpendicular 

 to the level surface in which the particle is contained ; but if the resultant be per- 

 pendicular to the level surface, the equilibrium will not be disturbed, because the 

 thin strata will still continue to exert a constant pressure upon one another in like 

 manner as before the new forces were introduced *. However the additional forces 



/d^ _ ^\ (^ ^' ^ <^\ /^ ^' d ^ \ 

 \dx~ dxj' \dy ~ dyj' \Tz dz) 



* It is by means of this very general principle that we pass from the equilibrium of a homogeneous fluid to 

 that of one in which the density, being constant at all the points of the same level surface, varies, according to 

 any law, from one level surface to another. 



