Theory of Capillarity . 313 
Laplace employs a function yjr, such that 
$t r 3 <p(r)dr=\ f(r)dr; .... (6) 
Jo Jo 
and he finds that in the case of a uniform fluid in contact with 
air the principal term, K, depends upon \^r{r) dr, and the 
second, H, upon ^r^r^dr. For the continuously varying 
fluid here considered, we see from (5) that 
p 1 -p 2 =27r(pl-pl)( ^(v)dr, ... (7) 
«- 
and that there is no term of the order of the capillary force. 
Equation (7) agrees with our general result that the difference 
of pressures required to equilibrate the forces operating be- 
tween two points depends only upon the nature of the fluid at 
the final points; and it shows further that, under the more 
special suppositions upon which the present calculation pro- 
ceeds, the molecular pressure at any point is to be regarded as 
proportional to the square of the density. 
But what is more particularly to be noticed is that, in spite 
of the curvature of the strata, there is no variation of pressure 
of the nature of the capillary force; from which we may infer 
that the existence of a capillary force is connected with sud- 
denness of transition from one medium to another, and that it 
may disappear altogether when the transition is sufficiently 
gradual. 
For the further elucidation of this question we will now 
consider the problem of an abrupt transition. It does not 
appear that Laplace has anywhere investigated the forces 
operative at the common surface of two fluids of finite density, 
but the results given by him for a single fluid are easily ex- 
tended. 
Let A (equal to a) be the 
radius of a spherical mass of 
liquid of " density " p 2 , sur- 
rounded by an indefinite quan- 
tity of other fluid of density p 1} 
and let us consider the varia- 
tion of pressure along a line from a point (say 0) removed 
from the surface on one side to a point B also removed from 
the surface on the other side. The difference of pressure cor- 
responding to each element of the path B is found by mul- 
tiplying the length of the element by the local density of the 
fluid and by the resultant attraction at the point. 
The attraction of the whole mass of fluid may be regarded 
