CAPILLARY ACTION. 561 



where A is a constant. But by the equations of equilibrium of the liquid 



dp= -pd x (2). 



Hence pdx = 2Apdp (3), 



and x'-X = 2 ^p-2 (4), 



where B is another constant. 



Near the plane surface of a liquid we may assume p a function of z. We 

 have then for the value of x a ^ the point where z = c, 



r'p(z)t(z-c)dz (5), 



Jc-e 



where c is the range beyond which the attraction of a mass of liquid bounded 

 by a plane surface becomes insensible. The value of x depends, therefore, on 

 those values only of p which correspond to strata for which z is nearly equal 

 to c. We may, therefore, expand p in terms of z c, or writing * for z c, 



-) + + &c 



where the suffix (c) denotes that in the quantity to which it is applied after 

 differentiation, z is to be made equal to c. We may now write 



[+e 



The function // (x) has equal values for + x and - x. Hence afi/> (x) dx 



J - 



vanishes if n is odd. 



But if we write 



f+t i [+ 



K=tr\ $(x)dx, L = -TT\ y?ty(x}dx, 



1 f + ' 



M = . TT x 4 ^ (x) dx, &c. 



1 . / . ) . 4 J 



This is the expression for x on ^ ne hypothesis that the value of p can 

 be expanded in a series of powers of z c within the limits z c and z + e. It 



VOL. II. 71 



