116 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



the appropriate function ^exists in the original 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 L we obtain 



But now the values of ip\ and y 2 are no longer zero at the new 

 boundary, but we have there, approximately 



*- $ SN 



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 the new boundaries. Since according 

 to the general laws of potential functions we have 



8"L=-8 1 f^diPidssz f^S'hds 



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



sn=- * is 



we obtain the final value: 



**=**+*■*=-* jl*($y-*($y]**s. . <2e) 



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

 unchanged during the variation, therefore it is necessary that 



f*N*=0 (2/) 



Hence results the variation, 



= -/ ds SN\p 2 - 2h-} . (2g) 



Here p 2 and ^ designate the fluid pressure on the upper and lower 

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

 hydrostatic equations. Since p 2 and p^ contain arbitrary additive con- 

 stants c can be omitted. 



