Theory of Surface Forces. 297 



and if r be very small, 



in agreement with (15) . 



In a recent memoir* Fuchs investigates a second approxi- 

 mation to the tension of curved surfaces, according to which 

 the pressure in a cavity would consist of two terms ; the first 

 (as usual) directly as the curvature, the second subtractive, 

 and proportional to the cube of the curvature. This conclu- 

 sion does not appear to harmonize with (27), (29), which 

 moreover claim to be exact expressions. It may be remarked 

 that when the tension depends upon the curvature, it can no 

 longer be identified with the work required to generate a unit 

 surface. Indeed the conception of surface-tension appears to 

 be appropriate only when the range is negligible in comparison 

 with the radius of curvature. 



The work required to generate a spherical cavity of radius r 

 is of course readily found in any particular case. It is ex- 

 pressed by the integral 



p.4;Trr 2 .dr (30) 



J. 



As a second example we may consider Young's supposition, 

 viz. that the force is unity from to a, and then altogether 

 ceases. In this case by (18), II(/) absolutely vanishes, if 

 f> a ; so that if the diameter of the cavity at all exceed a, 

 the internal pressure is given rigorously by 



?=^j/W)<Hx^. • • (3i) 



When, on the other hand, 2r < a, we have 



P=7 fW)/V/+27r \\a-f)fdf 



Jo Jlr 



^{I'-t^+H- (32) 



coinciding with (31) when 2r = a. If r = 0, we fall back 

 upon K = 7ra 4 /6. 



We will now calculate by (30) the work required to form a 

 cavity of radius equal to ^ a. We have 



, p a 2 , 7rV/l , 1\ 



The work that would be necessary to form the same cavity, 

 * Wien. Ber. Bd. xcviii. Abth. II. a, Mai 1889. 



