342 
investigations, the relative lengths of the radius and width of 
semicylinder may be proportioned so as to overcome the effect 
of difference of expansion of the material of the float and water. 
In Fig. 2, let Bpc be a sector of a hollow metallic cylindrical 
~] 
A 
xX B iPaek 
7 
1 
1 
1 
\ 
\ 
Fig. 2. 
float; the triangle pce represents the volume of water pressing 
against the side pc, supposed as the force concentrated at c. 
The rotating force acting on the axis D is the area of the 
triangle pcE, multiplied by pe, the distance of cg from p, 
DC? 
or — 
column of water expanded, to x; the triangle pxz represents 
the volume of water, its force concentrated at the point ec’, to 
which c had been expanded ; and the rotating force now acting 
x pc. Let us suppose the side of the float, and the 
- Dx’ : 
at G, is = * DG, and as the point P also became p’ by expan- 
sion (equal ratio with e’), the balance is maintained; but 
water expands more than metals by equal increments of heat. 
Let represent the point to which the water expanded, and 
the triangle puF its volume, which is greater than pxz, but 
only equal to it in weight; the rarefied volume of water pxz, 
acting on Dx’, will not balance the weight acting at P’; but, 
to keep the semicylinder in equilibrium during changes of 
temperatnre, the breadth must be so much longer than the 
