REPORTS 
ON 
THE STATE OF SCIENCE. 
{ 
Report on Recent Researches in Hydrodynamics. 
By G.G. Sroxss, M.A., Fellow of Pembroke College, Cambridge. 
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 Exercices of M. Cauchy *, a method which depends on the considera- 
tion of the forces acting on a tetrahedron of the fluid, the dimensions of which 
are in the end supposed to vanish. This proof applies equally to fluids at 
rest and fluids in motion; and thus the hypothesis above-mentioned may be 
considered as the fundamental hypothesis of the ordinary theory of hydro- 
_ dynamics, as well as hydrostatics. This hypothesis is fully confirmed by ex- 
| periment in the case of the equilibrium of fluids ; but the comparison of theory 
and experiment is by no means so easy in the case of their motion, on account 
_ of the mathematical difficulty of treating the equations of motion. Still 
_ enough has been done to show that the ordinary equations will suffice for 
_ the explanation of a great variety of phenomena; while there are others the 
laws of which depend on a tangential force, which ts neglected in the common 
theory, and in consequence of which the pressure is different in different 
directions about the same point. The linear motion of fluids in uniform 
_ pipes and canals is a simple instance. In the following report I shall first 
- consider the common theory of hydrodynamics, and then notice some theo- 
ries which take account of the inequality of pressure in different directions. 
‘It will be cunvenient to consider the subject under the following heads :— 
__ I. General theorems connected with the ordinary equations of fluid motion. 
II. Theory of waves, including tides. 
* Tom. ii. p. 42. 
1846. B 
regen 
