and for the tension of a liquid of density σ we have
T = πσ² ∫∞0 θ(z) dz. (33)
The form of this expression may be modified by integration by parts. For
| ∫θ(z) dz = θ(z)·z − ∫z | dθ(z) | dz = θ(z)·z + ∫zψ(z) dz. |
| dz |
Since θ(0) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply
∫∞0 θ(z) dz = ∫∞0 zψ(z) dz, (34)
and
T = πσ² ∫∞0 zψ(z) dz. (35)
In Laplace’s notation the second member of (34), multiplied by 2π, is represented by H.
As Laplace has shown, the values for K and T may also be expressed in terms of the function φ, with which we started. Integrating by parts, we get