REPORT— 1847 . 
body reduces itself to the geometric problem : to separate from a body by a jil»> 
er am volume m such a manner that its centre of gravity maybewitlitliitrf* 
am volume m such a manner that its centre of gravity maybewitbtliitrf* 
w de body m one straight line, perpendicular to the plane which is to be dim 
L or brevity we shall call the centre of cravity of the whole bodvbriefly 
srantv. and that of thr. . r At. ... ....... i r_ .,:iw 
L or brevity we shall call the centre of gravity of the whole bodvbriefly 
Sranty. and that of the «c>]»arat«i volume Me volume's centre.] For eqoilibn* 
positions where the ^aerating lines of cylinders or prisms are borizonUl.tber 
Diem iso! two dimennuns, and reduced to this: to separate from the area coitBS 
withm the curve A B d D. tn- a straight line AC, 
such an area ADC, that iu centre of gravity f 
may be with that of Uic whole area 0 ou a line 
^ perpendicular to A C. Let the equation of 
^ ® ^ ^ of A C 
y ar+b, i « • x, y the coordinates of the 
^of the centre nf gra- 
Mty ot the area ADC; and finally « the area 
A D C, we have then 
—Off— 
(af|+«'o)tfB+2(/5=0, 
di 
““y point of the curve formed by the volume’s 
thL iT I* corresponds to this point. The solution of 
tohm.the centre nf gravity to 
M ‘be centre of gravity of the wbri 
of the separated part, and is perpendicular to the separating 1"^- 
‘>f .mtersection of the normal with the curve. Thu 
of the floating body. 
brium 7^' l'os«ble to draw from the centre of gravity, as “““7 P®* ti,f 
F A D C be one of the second order, the 
»atM A ’^t'll na that which involves the scparnling i**’*^' AC*"* 
HMoud order, anti of the snme nature as the first. If the separatiug '‘f» ^ 
U.i? ti!"® straight Hues, then the curve of the volume’s centre j^| ^ 
thi the sides of the angle. For ** 
vcihinio' ” ® 'Hie radius of curvature in any point of jj„t 
J^llL • ‘2“- 'Fhe eentre e“rve» ^ irfs*- 
ir. “? ‘bemselves, and in none of their points is the radius of cu» 
or undetermined. ‘ rtbe*^ 
“"f l'.”r *= i." ‘ta' dtaensions. let ^ TlTjl^Klr./J 
of the 
volume 
volume, then is 
, j'’'* prooiem in three dunensions, let toe „iwKlr»r 
0 y be z_./(jr, y), that of the plane which separates from ^jjy d 
; " b® f=a®+ty+c. ,,, ( the coordinates of the centre of f» 
-~ax—hy—c]xdxdy, 
i)~ta—hy—c\ydxdy. 
y))“-Cflir+iy+c)3<irdy, 
