# Two questions about Gauss's contributions to capillarity and the calculus of variations

• In the last page of the abstract of Gauss's paper on capillarity "Principia generalia theoriae figurae fluidorum in statu aequilibrii" (1829), the author (who is he?) mentions (Gauss's werke, volume V, p.292) that Gauss's equation for the free surface of a liquid agrees fully with Laplace's equation (Laplace discovered that a free liquid surface in a cylindrical container assumes the shape of a surface of rotation of "elastica"). But in addition, he makes the following remark:

...secondly, for the boundaries of this surface, in the equation $$sin \frac{1}{2}i = \frac{\delta}{\alpha}$$, if $$i$$ denotes the angle of inclination between the planes touching the free surface of the liquid and the surface of the vessel, measured outside the liquid just the second main sentence used by Laplace without justification.

This passage doesn't have correct grammer (i used google translate), but i copied it as it was to prevent misunderstandings. This formula isn't clear to me; the verbal description by the author seems to indicate it's connected with the mathematical condition for "contact angle" (wetting angle), but the formula for contact angle is different than this. So my question is about the meaning of this formula; the author describes it as "Laplace's second main sentence", so perhaps someone familiar with Laplace's capillary work can answer.

• My second question is less important, and deals with a very small contribution of Gauss to the calculus of variations and elastica theory. In a fragment from volume 12 (p.49-52), Gauss deals with three problems of the calculus of variations; one of them concerns the derivation of the shape of "elastic curve", a problem that Euler worked about very intensively.

According to Oscar Bolza, Gauss offered an alternative solution to the same problem of variation "which is unlikely to be surpassed in elegance". The basic variational principle of the theory (the so called Euler-Bernoulli theory) is that the shape of the elastica minimizes the integral of the squared curvature, and Gauss solves the problem of shape of elastica that is tangent to two given straight lines at it's end points.

Gauss's derivation seems to be altogether very short and uncomplicated, so it doesn't look like a significant aspect of his work, but Bolza states the same method was applied by Max Born in his 1906 work on elastic curves in the plane and space. So this question is about explanation of what Gauss has actually done in this note.

• I finally found Laplace's "theory of capillary attraction" on internet archive and i must confess that i was amazed by the depth and thoroughness of his investigations - he deals with so many classical problems... from looking at laplace's treatise i think that since Laplace dealt with some problems of the shape of liquid trapped between inclined planes, perhaps Gauss is also concerned with this problem - maybe $i$ is just the angle of inclination between the planes and it's not connected with the contact angle. – user2554 Aug 12 '19 at 0:31