470 On the Theory of Surface Forces. 



The tension of the composite surface is thus given by (6) . 



If a = 0, we fall back upon the case of a simple sudden 

 transition from p 2 to p l} and we get as before 



Again, if a = qo , 



T=*(p 2 - Pl ) 2 f>(?)?sr. 



T=^{(p 2 -/>) 2 +(p-Pi) 2 }J>(?)?^. 



(') 



(8) 



This corresponds to the formation of two independently acting 

 tensions between the two pairs of liquids. 



To pass from these verifications to circumstances of novelty, 

 let us now suppose that a is small compared with the range of 

 the forces. When f is small, yfr(%) may be identified with 

 i/r(0), and we have 



hT=-Tr(p- Pl )( P2 -p).f(0).*% . . . (9) 



showing that in the limit ST is proportional to the square of 

 the thickness a. 



According to Young's supposition I. (19) * of a constant 

 attraction within the range a, 



*(0 = ia(a«-F) -*<«?-?). 



so that yfr[0)= ^a 3 ; and more generally whether a be great 

 or small, 



i 



f(?)^r=« 2 ( 1 v« 3 -i«« 2 + T 1 5« 3 )- • • (io) 



The general formula (6) may be applied also to the case of 

 a thin lamina by supposing that p 2 = p l = p . Thus 



T=2^(p-p ) 2 j o f(?)r^r- • . . (ii) 



gives the tension of a lamina of densit} r p and thickness a 

 surrounded by fluid of density p f . Here again, if a be very 

 small, the integral reduces to \a 2 ^(0), so that the tension varies 

 as the square of a. 



It must be understood that the lamina is here supposed to 

 be of uniform constitution, and that thus the result is probably 

 inapplicable to soap-films. 



* Phil. Mag. Oct. 1890. 



t In Maxwell's solution of this problem, Art. " Capillary Action/' 

 Enc. Brit., the tension of the lamina is given at double the above value. 



