Theory of Surface Forces. 291 



mass and to be gradually expanded in such a shape that the 

 walls consist almost entirely of two parallel planes. The dis- 

 tance between the planes is supposed to be very small compared 

 with their ultimate diameters, but at the same time large 

 enough to exceed the range of the attractive forces. The 

 work required to produce this crevasse is twice the pro- 

 duct of the tension and the area of one of the faces. If we 

 now suppose the crevasse produced by direct separation of its 

 walls, the work necessary must be the same as before, the 

 initial and final configurations being identical ; and we recog- 

 nize that the tension may be measured by half the work that 

 must be done per unit of area against the mutual attraction 

 in order to separate the two portions which lie upon opposite 

 sides of an ideal plane to a distance from one another which 

 is outside the range of the forces. It only remains to calcu- 

 late this work. 



If ct v cr 2 represent the densities of the two infinite solids, 

 their mutual attraction at distance z is per unit of area 



8wi«r,j" V(«)<fe, (5) 



or 27rcr 1 cr 2 6(z\ if we write 



1 



f(z)dz=6(z) (tf) 



The work required to produce the separation in question is 

 thus 



2w i< r,f*fl(*)«fc; ( 7 ) 



Jo 

 and for the tension of a liquid of density a we have 



T=iiwM" 0(*)d* (8) 



The form of this expression may be modified by integration 

 by parts. For 



$0(z) dz = 6{z).z-$z d6 ^dz = e(z) .z+Sz+iz) dz. 



Since 0(0) is finite, proportional to K, the integrated term 

 vanishes at both limits, and we have simply 



r* oo /» oo 



0{z)dz=\ z^(z)dz, .... (9) 

 Jo Jo 



and 



T 



= 7ra 2 \z^{z)dz (10) 



Jo 



