4 
metrical sphere, we are restricted to 
systems of four, sis, eight, twelve, and 
twenty balls, eacli touching the others of the 
same system as follows : viz : — Three others in 
the system of four, four others in the system 
of six, three others in that of eight, five others 
in that of twelve, and three others in that of 
twenty. A case of fluid equilibrium which 
can only occur where the particles of the fluid 
do not exceed twenty in number, can scarcely 
be held to affect the fundamental 
principle of hydrostatics. And the fact 
that, while the number of regular polygons 
is unlimited, that of the regular polyhedra is 
limited destroys (except in the particular iu- 
stancesjust adverted to) the analogy between a 
line or circle of particles in equilibrium and a 
sphere of like particles in equilibrium* and 
prevents it from being urged in support of the 
new hydrostatical paradox. I do not, at 
present, call to mind any investigations in 
which a perfect continuity of the fluid is 
assumed, unless, probably, in some of those of 
Professor Challis of Cambridge. But, even if 
I am right in thinking that he has assumed it, 
ali the ends. that he had in view would pro- 
bably be equally well served by changing the 
assumption to that of particles or distances in- 
finitesimally small in comparison with the 
particles whose motion is discussed or the 
mutual distances of the latter particles. At 
all events, a hypothesis assumed for a special 
purpose ought not to influence the present dis- 
cussion, unless it explains phenomena to be 
explained in no other way. 
8. The objection here taken cannot be 
answered by any assumption short of that of 
the absolute continuity oHlie flood. It is not 
auswered by an hypothesis that a fluid consists 
of infinitesimally small discrete particles. The 
difficulty arising upon the geometry of the 
question holds for any actual magnitude of the 
discrete particetq however small, and to 
assume the infinite physical divisibility of the 
fluid, is to assume its absohite continuity. 
Nor is there anything in the researches of 
Poinsot and Cauchy (I speak from the notice 
of their investigations by Mr. Cayley in the 
Philosophical Magazine, ser. iv, vol. xvii, pp. 
123,209), as it seems to me, to rebut it. Even 
if the Poinsot’s four new regnlar poly hedra 
(in an extended signification of the term) could 
be employed in the above investigation in the 
same way as the regular polyhedrons of ordinary 
geometry, the conclusion would remain that 
Ostrogradsky’s theorem can only be true for a 
limited number of discrete particles. Nor can 
any objection be successfully taken on the 
ground that I have assumed that the particles are 
spherical. It is sufficient for me to have shown 
that on that supposition Ostrogradsky’s theorem 
does not hold, and, speaking of those who have 
preceded me in the discussion with the deference 
due to recognized learning andeminent scientific 
position, I say that is incumbent on him who 
controveits, not on him who defends, the re- 
ceived principles of hydrostatics, to show under 
what other circumstances the theorem can 
obtain. 
9. The resea ches of Ostrogradsky and of Mr. 
Walton are, as might be anticipated, mathe- 
matically speaking, of high interest. The only 
exception which I take is that the hypothesis 
of continuity which they tacitly involve is un- 
necessaiy. I might go further, and say that it is 
perhaps more contradictory to than consonant 
with current notions, wh cli appear rather to 
reg ird all matter as consisting of discrete mole- 
cules or particles th in as perfectly continuous. 
Without going into any lengthened inquiry on 
this point I would cite a paper of Dr. Waugh, 
read to this society, in which he speaks of “ ul- 
timate atoms.” Whether the constitution of 
fluids be atomic or not, a general theory of 
hydrostatics ought alike to embrace them. If 
it be atomic, then such a state of things as that 
contemplated in article 6 of this paper could by 
no possibility arise. Each particle at the 
under surface of the ocean could not be simi- 
larly situaled in respect to all the other par- 
tichs on that surface, and, from this lack of 
symraetery, however sudden, extensive, or uni- 
form might be the contraction of the earth, the 
superincumbent waters would rush down upon 
it, and, after reaching it, be not strictly in 
equilibrium, but subject to the internal cur- 
rents to which the necessarily un symmetrical 
distribution of the atoms would give 
rise. The nearest approach that could be 
made to the state of things pictured by 
Ostrogradsky is that while the mass of waters 
rushed towards the centre, there might possibly 
be left of each spherical layer two rings of 
atoms, intersecting at right angles, and lying 
along great circles of the spherical surface. I 
say possibly, because it is just conceivable that 
the atoms of the whole mass might be symme- 
trically distributed in regard to such pairs of 
rings. But it is not worth while now to dis- 
cuss this phantom of the original theorem. 
10. I conclude therefore that the funda- 
mental principle of hydrostatics, viz., the 
equality of fluid pressure at any point of a 
fluid in all directions, is unshaken, save in a 
certain case of the unstable equilibrium of a 
perfectly continuous liquid ; and that, in as 
much as the existence of such a liquid is 
hypothetical only, the exception to the un- 
iversality of the fundamental principles of 
practical hydrostatics is likewise hypothetical 
only, and I also conclude that, if a fluid con- 
sist of discrete particles, then an equilibrium, 
stable or unstable, in which the recognized 
laws of fluid pressure do not hold, is not even 
theoretically possible. 
Printod by G, Wight, “ Guardian Office,” Brisbane. 
