116 THE MECHANICS OF THE EARTHS ATMOSPHERE. 



the appioi)riate function //exists in tbeori<?iual boundary, therefore, es- 

 pecially when the boundary -line forms a sharp corner, is a continuous 

 prolongation of the function excluded. The special physical signifi- 

 cance of such a case we shall have to consider later on. 

 By this first step in the variation of i we obtain 



.. .=i/[. (;| )%(*)>.. ^. 



But now the values of //i and i/'2 aie no longer zero at the new 

 boundary, but we have there, approximately 



and in order again to make these equal to zero we must execute a 

 second step in the variation, such that the function //' shall so vary that 

 these now again become zero at tbe new boundaries. Since according 

 to the general laws of potential functions we have 



S"L=-s, J jp^ d,/^, ds-s, j ^p 6^2 ds 



therefore when we (as is necessary in our case) put 



we obtain the final value : 



6L = S'L + S"L=-i f[s: {jPJ-^, (jPJ] <h SS. . (LV) 



Since finally tbe volume of each of the two liquids must remain 

 unchanged during the variation, therefore it is necessary that 



f d N ds=0 (2/) 



Hence results the variation, 



,{^-l} =-f dssis^\, (.,-..) -. 4- |(;^j- 1 (^)V. } 



= -/f7sr5xY|j>2 -pi.J (2^') 



Here pi and ;)] designate the fluid pressure on the upper and lower 

 sides, respectively, of the boundary surface as they result from Euler's 

 hydrostatic equations. Since jjg and j?] contain arbitrary additive con- 

 stants c can be omitted. 



