SURFACES OF DISCONTINUITY 599 



When we take gravity into account p, c, ni, m2 are no longer 

 constant throughout a given homogeneous mass or on a given 

 dividing surface; they are now functions of position as well as of 

 7], V, s, Wi, W2, and vary in value from point to point in one phase. 

 Considering a given infinitesimal element of volume Dv in the 

 unvaried state, it will change in size to Dv + ^Dv and will 

 move to a new adjacent position; the value of p at its new 

 location will vary not only for the reasons which would cause 

 variation without influence of gravity and which have been 

 involved in the earlier treatment of equilibrium, but also because 

 it has moved to a different position; and the veriest tyro in 

 hydrostatics knows that if a difference of level is produced pres- 

 sure will vary. This is where great care is needed; when 

 p8Dv is written in [600], the 8Dv is multiplied by the pressure 

 which existed where the element Dv was situated before the varia- 

 tion was conceived to take place. A similar remark applies to 

 fynbDmi^, fcdDs, fixiWmi^, etc. Now the term fpWv 

 should be written 



fp'bDv' + fp"bDv", (23) 



and considering the first integral in this we can regard it as the 

 sum of two parts, one in which the varied positions of the Dv' 

 still remain in the volume occupied by the single accent phase 

 before variation, and one in which the varied positions of the 

 Dv' are to be found between the original and varied situations 

 of the dividing surface. To evaluate the first part we shall for 

 the moment represent the element Dv' before variation by 

 Dva and after by Dvh, Dva being situated at x, y, z and Dvb at 

 X -\- 8x,y -\- 8y,z -{- 8z. The value of the first part is equal to 



fpa'iDvb' - DvJ) = fp^Dvi' - fpa'DvJ. 



But since the extent of integration is the same for the second of 

 these as for the first we can write this equal to 



fp^Dv,' - fpiIDVi! = /{pa' - Pb')DVi' 



= f{v'{^i y, 2) - p'ix -^ 8x,y -h 8y, z + 8z)]Dv' 



/I' 



' ^ aa; dy ^ dz J 



