where ρ is the density of the liquid, or if there are two fluids the excess of the density of the lower fluid over that of the upper one.
The forces acting on the portion of liquid P1P2A2A1 are—first, the horizontal pressures, −½ρgy1² and ½ρgy2²; second, the surface-tension T acting at P1 and P2 in directions inclined θ1 and θ2 to the horizon. Resolving horizontally we find—
T(cosθ2 − cosθ1) + ½gρ(y2² − y1²) = 0,
whence
| cosθ2 = cosθ1 + | gρy1² | − | gρy2² | , |
| 2T | 2T |
or if we suppose P1 fixed and P2 variable, we may write
cosθ = constant − ½gρy² / T.
This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.
| Fig. 8. |
Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sinθ2 − sinθ1), and this is therefore equal to the area P1P2A2A1 multiplied by gρ. The form of the capillary surface is identical with that of the “elastic curve,” or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and opposite forces applied either to the ends themselves or to solid pieces attached to them. Drawings of the different forms of the curve may be found in Thomson and Tait’s Natural Philosophy, vol. i. p. 455.