TRANSACTlOlSrS OP THE SECTIONS. 
3 
»btrt the double integrals refer to all elements of the area of the body’s intersection 
with the plane z=ax+hy-\-c. If we change the position of this plane m such a 
naner v remains constant, we obtain 
daJJ‘xds(iy-\-dhlJydxdy -h dc^dxdy^^, 
x 4 i=.~dajja^dx(ly ~d}^f*’xyfVsdy-~dcj'jxdxdy, 
tdr,= -dnJJ'xydxily — dh^fdxdy-dijjydxdy. +6dij. 
The last of these equations ajiows that the tangent plane in any point to the 
"Mre’s surface is pwallel to a cutting plane corresponding with, this point, so that the 
•IsKtion of the normal drawn from tire centre of gravity to the surface of the wnfr®s 
*^ta the position of equilibrinra of the floating body. If the body be limited 
V*"rftce of the second order, then the surface of the centres will also be of the 
•*•<1 degree, and of the same nature as that of the body. For a triangular py- 
**>1 the eqoation of the centre's surface is an algebraic of the third degree, for a 
tr-uguUr prism it represents a parabolic hyperboloid. If the cutting plane ro- 
abuat any axis through the centre of gravity of its area, then the volume s centre 
a cum of which the radius of curvature where 
^ ■ the tnotnent of inertia of the cutting area about the axis of rotation, and 
^Jf^dxdy. [The origin of the coordinates is taken In the centre of gravity of 
area, and the axis of rotation as that of the ».] The surface of the 
* fentre is a limited one, and nowhere can its radius of curvature be infinite 
•Wetennined. 
"elliptic cylinder can have in general eight equilibrium-positions in which the 
I'M is horizontal. If the centre of gravity be in one of the axes of the 
<aa Am*"* general six positionfl for the equilibrium ; but there are cer- 
Ataiih of the cootie of gravity from the centre of tho ellipse for which the 
are only two. If the centa* of gravity coincides with tho centre 
*r**’ equilibrium-positions. A cylinder, the directrix of which 
M’prrboiic curve and a straight line, ancl which, floats in such a man- 
^ fftnr without the fluid and the generating line horizontal, can 
; for Ae parabolic curve there arc only three. A 
^ 6w ^ centre of gravity coincides with that of its volume, can 
^ when tho whole surface is imraerged ; but there we 
^ 1 Y. •^!‘'*^*positiona for every one of its angles immerged or without the 
M general twelve equilibrium-positions. If the centre 
tnen are only two ijositionB possible, except the case 
sifintn^r* 1 '** rotation. If the centre of gravity coincides with the centre 
p oquilibriom-positions. 
w«. ’ Y ^ 9 ’*di 0 rium of Fhaiiug Bftdifij.— This tlieory contains both cases 
^ ftf und that of continuous mass. The small motion about 
futirtl. ennilibrium is shown to be represented by elliptic and uh{a- 
^ ^ pMs the general theory, and proceed directlv to the stability 
®’ The Complete moment of the forces acting on tne iloatiug body 
Z-^y^zdxdy 4- 
'"^‘*'~J]*H^=dxdz. 
(A.) 
^ 2 '^ whole mass of the floating body, the double only 
?rtvitr r ^ denotes an arbitrary infinite small displacement of the 
tw„if direction; and S\tr.j are infinite small angles of 
^^"ctionofiK which are perpendicular to one another; ff de- 
gravity; 5 the density of the fluid; dm the element of the body s 
B 2 
