ON RECENT RESEARCHES IN HYDRODYNAMICS. 17 
_ than they do when they are at rest*. The alteration of attraction or re- 
 pulsion is supposed to be, for a given distance, proportional to the velocity 
with which the molecules recede from, or approach to each other; so that 
the mutual repulsion of two molecules will be represented by f(r) — VF (r); 
where r is the distance of the molecules, V the velocity with which they recede 
from each other, and f(r), F (7) two unknown functions of 7 depending on 
the molecular force, and as such becoming insensible when 7 has become 
sensible. This expression does not suppose the molecules to be necessarily 
receding from each other, nor their mutual action to be necessarily repulsive, 
since V and F (7) may be positive or negative. It is not absolutely necessary 
that f(7) and F (7) should always have the same sign. In forming the equa- 
tions of motion M. Navier adopts the hypothesis of a symmetrical arrangement 
of the particles, or at least, which leads to the same result, neglects the irre- 
gular part of the mutual action of neighbouring molecules. The equations 
at which he arrives are those which would be obtained from the common 
@u du du 7 f = h 
(Gatgetae) in place of 5 in the 
first, and making similar changes in the second and third. AS is nae an 
_ unknown constant depending on the nature of the fluid. 
The same subject has been treated on by Poisson, who has adopted hy- 
potheses which are very different from those of M. Navier. Poisson's theory 
is of this nature. He supposes the time ¢ to be divided into 2 equal parts, 
each equal to 7. In the first of these he supposes the fluid to be displaced 
in the same manner as an elastic solid, so that the pressures in different 
_ directions are given by the equations which he had previously obtained for 
elastic solids. If the causes producing the displacement were now to cease 
to act, the molecules would very rapidly assume a new arrangement, which 
would render the pressure equal in all directions, and while this re-arrange- 
ment was going on, the pressure would alter in an unknown manner from 
that belonging to a displaced elastic solid to the pressure belonging to the 
fluid in its new state. The causes of displacement are however going on 
_ during the second interval 7; but since these different small motions will 
_ take place independently, the new displacement which will take place in the 
second interval 7 will be the same as if the molecules were not undergoing a 
: " re-arrangement. Supposing now z to become infinite, we pass to the case in 
which the fluid is continually beginning to be displaced like an elastic solid, 
| continually re-arranging itself so as to make the pressure equal in all 
directions. The equations at which Poisson arrived are, in the cases of a 
romogencous incompressible fluid, and of an elastic fluid in which the change 
of density is small, those which would be derived from the common equations” 
d 
equations by writing ~ — 
d 
| py replacing 7’ <P iin the first by 
dp _ du ald i se lh es 
dx dat dy? +o BE dy dz 
% and making similar changes in the second and third. In these equations a. 
and B are two unknown constants. It will be observed that Poisson’s equa- 
ions reduce themselves to Navier's in the case of an incompressible fluid. 
___ The same subject has been considered in a quite different point of view by 
™M. Barré de Saint-Venant, in a communication to the French Academy in 
* This idea appears to have been borrowed from Dubuat. See his Principes d’Hydrau- 
_ lique, tom. ii. p. 60. 
| Y Journal de l’Ecole Polytechnique, tom. xiii, cah. 20. p. 139. 
| 
