CAPILLARY ACTION. 577 



from the pressure p over the area of the section. In the next place, there is 

 the surface-tension acting downwards, but at an angle a with the vertical, 

 across the circular section of the bubble itself, whose circumference is 2iry, and 

 the downward force is therefore 



Now these forces are balanced by the external force which acts on the 

 disk ACB, which we may call F. Hence equating the forces which act on 

 the portion included between ACB and PRQ 



Try*p-2TryTcosa = -F ............................. (9). 



If we make CR z, and suppose z to vary, the shape of the bubble of course 

 remaining the same, the values of y and of a will change, but the other 

 quantities will be constant. In studying these variations we may if we please 

 take as our independent variable the length s of the meridian section AP 

 reckoned from A. Differentiating equation 9 with respect to s we obtain, after 

 dividing by 2ir as a common factor 



Now -^ = sina .................................... (11). 



The radius of curvature of the meridian section is 



The radius of curvature of a normal section of the surface at right angles to 

 the meridian section is equal to the part of the normal cut off by the axis, 

 which is 



(13). 



cos a 

 Hence dividing equation 10 by 7/sina, we find 



This equation, which gives the pressure in terms of the principal radii of curva- 

 ture, though here proved only in the case of a surface of revolution, must be 

 true of all surfaces. For the curvature of any surface at a given point may 



73 



VOL. II. 



