j ( 233 ) 
[April 1, 
PROCEEDINGS OF LEARNED SOCIETIES. 
= 
: NATIONAL INSTITUTE OF FRANCE. 
LAPLACE lately read a memoir 
e on the apparent attraction and 
repulsion of small bodies swimming on 
the surface of fluids. 
In this theory of capillary action, M. 
Laplace had submitted to analysis the 
attraction which takes place between 
two vertical and two parallel planes, 
nearly brought in contact with each other, 
and -having their inferior extremities | 
plunged into a fluid. He demonstrated 
that, if they were composed of the same 
material, thisaction tended to make them 
approach each other, whether these 
planes lay on the surface of the fluid, as 
when plates of ivory are plunged into 
water, or whether they sunk, as plates of 
laminated talc, having an unctuous touch, 
which prevents them from becoming 
moistened. Fach -plane experiences 
then towards the other, a pressure equal 
to the weight of a parallelopiped of the 
same fluid, of which the height will be 
the half sum of the elevations above the 
level, or of the sinkings below it, of the 
extreme points of contact, of the exte- 
rior and interior surfaces, of the fluid with 
the plane, and cf which the base will 
form that part of the plane comprehend- 
ed between the two horizontal lines run- 
ning through these points. This theorem 
contains the true cause of the apparent 
attraction between bodies swimming on 
the surface of a tiuid, when it 1s elevated 
or sunk near them. But experiments 
have shewn that bodies repel each other 
when the fluid is elevated towards the 
one while that it sinks towards the 
other. 
M. Laplace having applied his analysis 
to these repuisions, was led to the fol- 
lowing results, which must prove inter- 
esting to the geometrician, and com- 
plete the theory of capillary action. 
If we always suppose that bodies, are 
vertical and parallel planes, the section 
of the surface of the fluid comprised be- 
tween them, by.a plane vertical and per- 
pendicular to these planes, has a point 
of inflection, while they remain at the 
distance of some centimetres from each 
other, In proportion as they approach, 
the point of inflection approaches from 
the plane near to which the fluid sinks, 
if the sinking of the fluid in contact with 
the exterior of this plane, is less than 
the elevation of the fluid in contact with 
the exterior of the other plane. In the 
contrary case, the point of iniection ap- 
proaches from this last plane. This 
point is always on a level witi the fluid 
in the vase into which the planes are 
plunged. ‘The elevation and sinking of 
the fluid in contact with these planes, 
are less at the interior than the exterior. 
In this case the two planes repel each 
other. As they approach, the repulsion 
still continues, while the point of imflec- 
tion remains. This point terminates at 
length by coincidmg with one of the - 
planes. The repulsion still, however, 
continues even beyond this point; but 
on the planes continuing to approach it, 
ceases, and is ultimately converted into 
attraction, ¢ this instant, the fluid is 
equally elevated at the interior and exte- 
rior of the plane susceptible of being 
moistened ; it is as much elevated above 
the level of the mterior of the plane, as~ 
it is sunk below the exterior. Thus 
the repulsion is changed ito attrac- 
tion in the same instant in both. the 
planes. On their near approach they 
attract each other, and unite by an acce- 
lerated motion. ‘These planes thus dis- 
play the remarkable phenomenon of an 
attraction at very small distances, which 
becomes changed into repulsion beyond 
a certaii pout ; a phenomexson which is 
also exhibited in the inflection of light, 
near the surface of bodies, as well as by 
electric and magnetic attracuons. There 
is, however, one case 1n which the planes 
repel each other, however small may be 
their mutaal distance, and that is when 
the fiuid sinks near to one of them, as 
much as it is elevated near to: the other, 
Then the surface of the fluid has a con- 
stant inflection in the middie of the in- 
terval which separates, thetn. 
The integration of the differential 
equation of this surface depends in ge- 
neral on the rectification of the conic 
sections, and consequently it Is impossi- 
ble to obtain for it finite terms. But 
this becomes practicable whea the planes 
reach that point where repulsion is con- 
verted into attraction; then this distance 
can be ascertained from the elevation and 
sinking of the fluid at the exterior of the 
planes. - We find also that it 1s infinite, 
if the fluid sinks only a little at the exte- 
rior of the plane which 1s not susceptible 
of being moistened; whence it follows 
that in this case the two planes, cease to 
repel each other, This, however, can 
still take place, even when. the fluid sinks 
sensibly at the exterior of this last plane 
. generally 
