Manchester Memoirs, Vol. Ixv. (1921), No. \ 7 



analysis, second order corrections may be reached by the aid of very 

 simple mathematics, involving nothing more recondite than a few standard 

 differentiations and integrations. 



We shall take it for granted that, across any line of length ds drawn 

 in the surface of liquid there is exerted a tension Tds, whose direction is 

 normal to the line, and in the tangent plane to the surface. The quantity 

 T is called the Surface Tension of the liquid, and is reckoned, in C.G.S. 

 units, in dynes per centimetre (or grams per second per second). 1 Further, 

 if R x and R 2 are the principal radii of curvature at any point of a curved 

 surface in which there exists a surface tension T, then the pressure excess 

 on one side of the surface over that on the other side is given by 



This is the well-known equation whose solution has, from time to 

 time, engaged the attention of most of the famous analysts of the nine- 

 teenth century. Briefly, 3 we may summarise the results of their work in 

 the statements that (1) an exact solution of the equation in its most 

 general form cannot be attained ; (2) that even if we confine ourselves to 

 the case in which the surface is one of revolution we can obtain approxi- 

 mate solutions only in certain special cases ; and (3) that exact solutions 

 may be found in those cases where one of the principal radii of curvature 

 becomes infinite. This is the case contemplated when a liquid rises (or 

 is depressed) between two parallel plane surfaces held closely together, or 

 when a liquid is in contact with a plane vertical wall. 



The cases which come under the heading (2) are of most importance 

 from our point of view, and we shall study, in some little detail, the solu- 

 tion of the equation when the surface is one of revolution about a vertical 

 axis. Two special cases concern us most closely — (a) the shape of the 

 capillary surface inside a small vertical capillary tube, or, what amounts 

 to the same thing, the shape of the outline of a small pendent drop of 

 liquid, and (/3) the shape of the capillary surface formed when a large 

 bubble of air is imprisoned under the surface of a liquid or when a large 

 drop of mercury rests upon a solid surface. 



It is of the very first importance that we should attach clear ideas to 

 the terms large and small contained in the preceding sentence. 



Introducing the idea 4 of the "specific cohesion" a 2 , we may say that 



r 2 . . 



a small capillary tube is one in which -^ is small compared with unity. 



1 Not in dynes — an unfortunately common statement. 



2 For a simple proof, see Poynting and Thomson, " Properties of Matter," Chap. XIV. 



3 The discussion following is restricted to the capillarv surface under gravity, 



T 



4 a 2 is defined by the identity a 2 = — so that a has the dimensions of a length. As 



is well known, for liquids of zero contact angle, a first approximation to T is given by 

 T = %rhpg, where h is the height to which the liquid rises in a capillary tube of radius 

 r. Consequently 2a 2 is of the order rh. Some writers (jfour. Amer. Chem. Soc., 41, 

 520 (igig)) apparently use the equation 2a 2 = rh to define a 2 , and then deduce the 

 identity given above. This is quite incorrect. 



