Profile of a large drop

In class we derived the profile <math>z(x)</math> of a large drop as being:

<math>\sigma_{lv}(\frac{1}{\sqrt{1+\dot{z}^2}}-\cos\theta_e)=\frac{1}{2}\rho g(2ez-z^2)</math>,

where we had defined <math>e</math> as being the maximum height of the drop:

<math>e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}</math>,

and <math>\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}</math> was the inverse capillary length.