18 REPORT—1846. 
1843, an abstract of which is contained in the ‘Comptes Rendus*.’ The 
principal difficulty is to connect the oblique pressures in different directions 
du du 
about the same point with the differential coefficients Ts ’ ag &c., which 
express the relative motion of the fluid particles in the immediate neighbour- 
hood of that point. This the author accomplishes by assuming that the tan- 
gential force on any plane passing through the point in question is in the 
direction of the principal sliding (glissement) along that plane. The sliding 
along the plane wy is measured by Tz +5 in the direction of x, and 
d 
oy + 7 in the direction of y. These two slidings may be compounded 
into one, which will form the principal sliding along the plane ay. It is 
then shown, by means of M. Cauchy’s theorems connecting the pressures in 
different directions in any medium, that the tangential force on any plane 
passing through the point considered, resolved in any direction in that plane, 
is proportional to the sliding along that plane resolved in the same direction, 
so that if T represents the tangential force, referred to a unit of surface, and 
S the sliding, T=eS. The pressure on a plane in any direction is then 
found. ‘This pressure is compounded of a normal pressure, alike in all di- 
rections, and a variable oblique pressure, the expression for which contains 
the one unknown quantity ¢. If the fluid be supposed incompressible, and 
é constant, the equations which would be obtained by the method of M.Barré 
de Saint-Venant agree with those of M. Navier. It will be observed that 
this method does not require the consideration of ultimate molecules at all. 
When the motion of the fluid is very small, Poisson’s equations agree with 
those given by M. Cauchy for the motion of a solid entirely destitute of elas- 
ticity +, except that the latter do not contain the pressure p. These equations 
have been obtained by M. Cauchy without the consideration of molecules. 
His method would apply, with very little change, to the case of fluids. 
In a paper read last year before the Cambridge Philosophical Society}, I 
have arrived at the equations of motion in a different manner. The method 
employed in this paper does not necessarily require the consideration of ulti- ” 
mate molecules. Its principal feature consists in eliminating from the rela- 
tive motion of the fluid about any particular point the relative motion which 
corresponds to a certain motion of rotation, and examining the nature of the 
relative motion which remains. The equations finally adopted in the cases 
of a homogeneous incompressible fluid, and of an elastic fluid in which the 
change of density is small, agree with those of Poisson, provided we suppose 
in the latter A= 3B. It is shown that this relation between A and B may 
be obtained on Poisson’s own principles. 
The equations hitherto considered are those which must be satisfied at any 
point in the interior of the fluid mass; but there is hardly any instance of 
the practical application of the equations, in which we do not want to know 
also the particular conditions which must be satisfied at the surface of the 
fluid. With respect to a free surface there can be little doubt: the condi- 
tion is simply that there shall be no tangential force on a plane parallel to 
the surface, taken immediately within the fluid. As to the case of a fluid in 
contact with a solid, the condition at which Navier arrived comes to this: 
that if we conceive a small plane drawn within the fluid parallel to the sur- 
* Tom. xvii. p. 1240. 
+ Exercices de Mathématiques, tom. iii. p. 187. 
t Transactions of the Cambridge Philosophical Society, yol. viii, p. 287. 
