10 PHYSICS. - 
preceding law, be inversely proportional tn the lengths, that is, one pound at 
the end of the longer arm will balance 4 01 10 pounds at that of the shorter. 
As the short arm, BF = C, is fixed, and the weight, W, subject to great 
variation, and as the counterpoise, P, is likewise constant, the arm, AF=D, 
must oe variable to hold any weight, W, in equilibrium. This is attained 
by shifting the point of suspension of the weight, P. Thus, let BF =1, 
A= 4, P=, then will P: W:: BF: AF, or 2:W::1:4; them W =18; 
and 2lbs. at A will balance 8 at B. If, however, W weigh less than 8lbs., 
then A hanging at P, the arm AF will preponderate, and P will have to be 
shifted towards the fulerum. Supposing equilibrium to occur at D, and that 
DF = 3, then we shall have the proportion 2: W ::1:3, and W will be equal 
to 6. This mode of calculation is, however, too tedious in practice, and 
therefore the long arm, AF, is previously graduated in such a manner, that 
when the weight and the counterpoise are in equilibrium, a number on the 
scale opposite the latter indicates the amount of the former. It is evident 
that the balance is accurate only so long as BF’, P, and FA, remain 
unchanged in length or weight. 
The law of the lever finds numerous applications in the determination of 
the centre of gravity. To obtain the centre of gravity of an irregular 
figure, as of the quadrilateral. ABCD (jig. 9), divide it by a diagonal into 
two triangles, determine by the preceding methods their centres of gravity, 
and consider the connecting line, SS’, of these centres. as a lever upon 
which, at S and S’, forces operate proportional to the surfaces of the two 
triangles. The centre of gravity or fulcrum, 8”, is obtained by dividing 
the line, SS’, in such a manner, that SS”: S/S”: : triangle BCD: triangle ABC. 
By continuing this process the same end may be attained for figures of 
more than four sides. The centre of gravity of two combined bodies, BE 
and be ( fig. 18), is obtained by uniting their separate centres of gravity, 
and dividing the connecting line, Ss, into two such parts at S’, that the 
distances of this point from the centres of gravity shall be inversely pro-- 
portional to the masses of the two bodies. 
If more than two forces act on one lever, striving to move it in two 
determinate and opposite directions, equilibrium occurs when the sum of 
the momenta of all the forces acting on one arm, is exactly equal to that of 
the forces operating upon the other arm. Thus in fig. 27 must P. AF+ 
PAF +P". A"F =W.BF + W’.B’F + W”.B’F. When the forees 
on the same arm of the lever operate in different directions, some upwards 
and others downwards, as in fig. 28, then equilibrium takes place when the 
difference of the momenta of the forces acting on one arm, is equal to the 
same difference in the momenta of the forces operating upon the other 
arm; thus, when W.AF—P.CF =P’. DF+ P”. EF—W’. BF. 
Fig. 31 represents a compound lever, consisting of three simple levers, 
AB, A’B’, A”’B”, acted upon in opposite directions by the weights P, W. 
) R.A. 
Upon the middle lever, whose fulcrum is I", the force Br. operates at A’, 
the force acting on B’ ule both of these forces press A’'B’ upwards, 
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