4 
REPORT— 1847. 
mass. The plane of ay is in the surface of the fluid, and the origin of cooidbifs 
in the centre of gravity of the area of intersection. 
If we change a little the position of the floating body so that the surfacerfii 
fluid may cut it in another direction, then the moment of the forces will also (Amp. 
l^e increasea of the double integrals which arise from the alteration of tiieliM 
disappear. If we have regard to the cotiditiuns of equilibrium and denote lirlf 
the inflnite small angle of rotation about the axis, in which the new section cotsde 
other, by M the moment of inertia of the cutting area about the same axis, by Sth 
area of Uie section, by p the immerged volume, b)* x and f the distances of thiw® 
gravity and that of the volume's centre from the cutting plane, then the rao*' 
or the expression (A.) takes the following form * 
for all displ^eraents 8y and 8^* which make this expression positive, the eqsS' 
riuni la labile, it is stable for the displacements 8y, dy' which give a wgatw 
for^is expression, if they reduce it to zero, then the equilibrium U neutral 
i ne application of these result.^ to the floating cylinder with its geaerehas® 
nnrtzontal shows, that ifthe centre of curvature in that pointof thecentrecQr«.*bi(i 
wrresponds to the equilibrium-position, lies higher ^on tlie centre of 
»*ti^*^r***?° ** stable! if it lies lower, the equilibrium is labile, or the eqoilibanas 
the distance of the centre of gravity from the centre curve isaroaiiss^ 
u It labile If this dmtanco is a minimum. Tlie equilibrium of any bodvissubliif* 
amailest radius of curvature in that point of the (centre surface, which carresposAJ 
iiic posiUori of equilibrium, ia greater than the distance of the same point &« » 
centre ol gravity. Ihe equilibrium is labile if this distance is greater tiiittue 
Tu" '■““'“5 curvature. The equilibrium can be neutral for all diieclioM oiilj. 
1^ . ® *^fnce of the centres oflera in the point, which corresponds to the f*!’*'*' 
I will yet cite some few of the applications. Al^' 
fSl, «‘‘l“>*‘hrium-po8iiions. If that of the middle 
traposaible i if Uiat of the middle is labile, then there ^ 
tic Afwi i.v'L r^ stable. Somewhat aimilar happens wtb ^ 
vir\- :» . cylinders. Au ellipsoid of rotation, whiidi has its centre ci?*’ 
tiMliv rotation, may Imvi*. besides the position in which this axis* 
not p 4 iblc when the first 
RhS f' nSli. ^ cJ’. POMible if the firat position ofl-ers a labile eqml*'®- 
have *1^ *=^ 0^31 heights and cliflerent regular polygon bases of 6*1“^,*^ 
amallor fli« bohcs, an equilibrium, which is the more 
basw ia on s'ties. Ilie equilibrium has the greatest stabititt>f 
is * circle. But if the basu»tf 
Saw S/r stability is smaller than in the case of thr 
oaau being a quadrate or on equilateral triangle*. 
On CaMm of QnalermonsIn th: TheomoJAtSfO- 
J . oir W. R. IIamutok, Astronomer Royal of Ireland. 
^‘^^***P^nf uHlst^rimetrimlPrM~in treatedhythe Calcidusof 
. ro . ir R. Hamilton, Astronotner Royal of lrcland. 
princip^*uxM of horizontal floats in eqnilibrinmwHb ^*y' 
which the same Bxia i* section vertical, there is a second position of equilU'Oj^ 
The author aim^r! ‘® '^‘>.'Tcsponding to the reversed position of the elliptic c>linJ 
the vertical thloZh ‘’t-crhiokcd the circumstance that this is no longer true ^ 
With «„e of the D?h.rin.t nv the position of equiUhrium does not ^ 
tionsiof,.q„jlj},j.jyP^ ^ P csof the section. Thus there cannot be more tbw fouf ^ 
“"c position of Duuilibriiiin cyhnrlcr floating with iu axisboriwntaJ, since lhcre «« 
Ol cquilibnun, corresponding to cac-h normal. {NotehySec^taryof Scrtii»«^ 
