341 
on pce will be diminished and the balance disturbed; the 
weight at Pp will preponderate and cause the float to sink, 
making the water rise again until it reaches BD, as before, 
when the weight and pressure will balance each other. The 
converse of this is also true, as, if the water be raised above BD, 
the pressure on pc will be increased, the float will rotate in 
the other direction, raising it out of the water, and lowering 
the surface level till it once more reaches Bp, and the balance 
will be restored, the water and the float remaining at rest. 
If the semi-cylinder be homogeneous, its centre of gravity 
will be distant from the axis p, the cube of the radius divided 
by one and a half times the area of the semicylinder, which, 
if the radius be considered as 1, is about 0°4244; the distance 
of p from p is 0°6666 = that of c. A weight at p’ (0°4244 
from D), to have an equal force with one at P’ (0°6666), must 
be inversely as their distances; therefore, the weight of the 
semicylinder, whose centre of gravity is at P’, is to the weight 
at P as 0°6666 to 0°4244. And, as the pressure at P, or G, is 
represented by the weight of a quantity of water equal in bulk 
to the triangle pcE, the specific gravity of the semicylinder 
will be inversely as its area to that of the triangle, and directly 
as their weights, or about one-half the specific gravity of the 
water. But if the weighting be applied at the circumference, 
as the centre of gravity of such an arc would be about the 
distance 0°6366 from p, multiplied by the radius, its agere- 
gate weight should be to the homogeneous semicylinder as 
0°4244 to 0-°6366, which would be nearly one-third lighter 
than if homogeneously balanced ; and that is the case, no mat- 
ter what the breadth of the float: but the same does not hold 
if the figure be a hemisphere, as the centre of gravity of such 
a body is about 0°375 from its axis, and the centre of gravity 
of a hemispherical surface is at five-tenths of the radius, which 
is greater than the ratio given for the semicylinder. 
The effect of expansion by heat is not very appreciable, 
however; if the semicylinder be employed for very accurate 
