2 
it, and approach it, too, to the same extent. 
But this they cannot do in such manner as 
that all the particles situate on any one and 
the same spherical surface described round the 
centre of attraction, slisll take the same 
movement ; for, if they could, a diminution of 
of the volume of the liquid would be the result. 
Thus, the liquid will remain equilibrium. 
But it is evident that the force which attracts 
each paiticle situate at the inner surface of the 
layer tends to urge the particle, not towards 
the liquid mass, but away from it.” 
Ostrogradsky then calculates the pressure at 
the inner surface, and says, “This pressure is 
certainly different from zero, which again is 
contrary to what has been heretofore admitted. 
Here, then, we have a singular case of 
equilibribrium which has escaped the known 
theory of liquids, and which authorises us in 
concluding that that theory lias not yet at- 
tained a befitting development.” 
3. Mr. Walton discusses this singular case of 
fluid equilibrium not only because it is in 
itself curious as a speculative question, but also 
because a right interpretation of such equi- 
librium — although, as being unstable, only 
theoretically possible— may tend to awaken 
criticism on the fundamental principles of 
practical hydrostatics. Mr. Walton then con- 
siders the question from Ostrogradsky’s point 
of view, observes that the pressure at the in- 
ternal surface need not exceed an infinitesimal 
quantity, and that the problem of internal 
pressures in Ostrogradaky’s shell is, iu fact, in- 
determinate ; and concludes that the theoretic 
demonstration of the equality of fluid pressure 
in all directions at any point is unsound in 
regard to unstable equilibrium, and that it is 
theoretically possible to have a complete sphere 
of fluid in equilibrium in which “ the principle 
of the equality of fluid pressure at any point in 
all directions” is not true, In this case Mr. 
Walton conceives that the fluid would be stable 
in regard to the geomet y of the fluid, and un- 
stable in regard to the relative mechanical 
actions of its elements. 
4. Now this supposed new hydrostatic al 
paradox not only explicitly assumes the abso- 
lute incompressibilty, but it also, as I shall 
show, tacitly implies the absolute continuity of 
the liquid. The explicit assumption is un- 
objectionable enough, for even if in nature 
there should exist no perfectly incompressible 
fluid, still the theoretical properties of such a 
fluid would, as an approximation at all events, 
guide us to the properties of a partially com- 
pressible fluid. And, more, the new paradox 
might have a place in the discus.-ion of the 
latter fluid, allowance being made for a 
contraction of the i iner surface before equi- 
librium is established. But the tacit assumption 
seems to me to be open to grave objection. It is 
superfluous. In order to establish tne science of 
hydrostatics, it is unnecessary to frame any 
hypothesis wliatsver as to the constitution of 
fluids, and, whether they be supposed to be 
absolutely continuous bodies or to consist of 
discrete solid particles, the conclusions of the 
science are alike valid. We sufficiently charac- 
terize a fluid when we say that its parts are 
freely moveable amongst each other without 
friction and on the application of the slightest 
force. The fundamental principle of hydro- 
statics, viz., the equality of fluid pressure at 
any point of a fluid in all directions, whether 
proved theoretically or confirmed experimen- 
tally, is true of a body so constituted. In 
fact, as we pass from the consideration of such 
bodies as water to that of oils and viscid 
matters, which require time to find their level, 
and, thence, to the case of powder, sand, or 
stones, which when heaped up find a sort of 
level though not a horizontal one, we see that 
the transition from ordinary mechanics to 
hydrostatics is not quite so abrupt as might 
be imagined. To make this plainer, conceive 
three billiard balls placed on a horizontal table 
in contact with each other, and a fourth to be 
placed on the top of and in contact with the 
other three. If we suppose the balls as well 
as the table to be perfectly smooth, and no 
friction or cohesion to act either between the 
balls themselves or between the balls and the 
table, the uppermost ball will descend, pushing 
the others apart and resting in the centre of 
the equilateral triangle, at the respective angles 
of which the other three will be found until 
they meet the edge of the table. The same 
thing would occur with any number of perfectly 
smooth balls placed anyhow on a perfectly 
smooth table. Were the table large enouga, 
each ball would ultimately descend to the 
table ; and so the system of balls would find a 
level, the counterpart of the hydrostatic level. 
The final state of a system of smooth balls 
placed anyhow on a smooth horizontal con- 
fined space would bs determined by the prin- 
ciple that the centre of gravity of the system 
would be as low as possible. 
5. At first sight the consideration of this case 
of perfectly smooth hard balls upon a perfectly 
smooth hard horizontal table seems to give a 
plausibility to Ostrogradsky’s paradox. If any 
number of such balls be placed in a straight 
line and in contact, then, whatever be the 
attractive force exerted by the balls, or what- 
ever equal external pressures be applied to the 
outermost balls, in the direction of the line 
joining their centres and towards the interior 
of the system, equilibrium, though unstable, 
will result. So if we place on the table a per- 
fectly smooth and hard circular cylinder, with 
its axis vertical, the relative sizes of the cylinder 
and the balls may be so adjusted as to render 
it possible to encircle the cylinder with a ring 
of balls, all of the same size, each of which shall 
be in contact with two contiguous balls, with 
the cylinder and with the table. This adjust- 
ment is conceivable, because we know that 
theoretically — that is to say without a geo- 
