At the meeting of the British Association held at Cambridge last year, the Committee of the Mathematical Section expressed a wish that a Report on Hydrodynamics should be prepared, in continuation of the reports which Prof. Challis had already presented to the Association on that subject. Prof. Challis having declined the task of preparing this report, in consequence of the pressure of other engagements, the Committee of the Association did me the honour to entrust it to me. In accordance with the wishes of the Committee, the object of the present report will be to notice researches in this subject subsequent to the date of the reports of Prof. Challis. It will sometimes however be convenient, for the sake of giving a connected view of certain branches of the subject, to refer briefly to earlier investigations.

The fundamental hypothesis on which the science of hydrostatics is based may be considered to be, that the mutual action of two adjacent portions of a fluid at rest is normal to the surface which separates them. The equality of pressure in all directions is not an independent hypothesis, but a necessary consequence of the former. This may be easily proved by the method given in the Exercises of M. Cauchy, a method which depends on the consideration of the forces acting on a tetrahedron of the fluid, the dimensions of which are in the end supposed to vanish.