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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 
float; the triangle pcz represents the volume of water pressing 
against the side pc, supposed as the force concentrated at G. 
The rotating force acting on the axis D is the area of the 
triangle pcr, multiplied by pe, the distance of c from bp, 
2 
or - x pe. Let us suppose the side of the float, and the 
column of water expanded, to x; the triangle pxz represents 
the volume of water, its force concentrated at the point 6’, to 
which c had been expanded ; and the rotating force now acting 
WAT ay 5 5 es ° j 
at G’, is > * pe’, and as the point P also became P’ by expan- 
sion (equal ratio with c’), the balance is maintained; but 
water expands more than metals by equal increments of heat. 
Let u represent the point to which the water expanded, and 
the triangle pur its volume, which is greater than pxz, but 
only equal to it in weight; the rarefied volume of water pxz, 
acting on px’, 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 
