MECHANICS. 9 
above, and the force, P, draws upwards, while the weight, W, draws down- 
wards. This form by some has, for this reason, veen called a lever of the 
third class. In this lever, the above-named conditions still hold good, and 
the same is the case in the bent lever (fig. 36). Here, however, the bend 
of the arms, A’F and FB’, of the lever, is not to be considered, but only the 
direct distances from the fulcrum, B’b and A/a, or the levers, AF and FB, 
equal and parallel to them. Here also in a state of equilibrium we have 
PeW BP uAP. 
Hitherto we have had reference to the mathematical lever, that is, to a 
line without weight; if the actual material lever be the one in question, 
where the weight of the arms of the lever comes into account, then the 
same proportions of the arms of the lever being retained, but with greater 
curvature of one or other arm, and consequently greater weight, the pro- 
portion, P and W, might change greatly without any disturbance of 
equilibrium. 
Considering closely the proportion P: W::BF: AF, we have P. AF = 
W. BF, this product of the two extremes and the two means being called the 
momentum of the forces. The momentum therefore of a force, is the product 
of the force by its leverage, and the preceding laws can be expressed in 
shorter phrase, by saying, a lever is in equilibrium when the momenta of 
the forces acting upon it are equal. 
The case is somewhat different when the forces acting on the lever are 
not parallel to each other, as in fig. 29, where the two forces, P and W, are 
carried over pulleys. In this case each of the two forces must be decom- 
posed into two others, of which one is perpendicular, and the other parallel 
to the lever. Expressing P by DA, and W by BG, calling also the angle, 
DAC, «, and the angle, GBE, §, then the force, DA, may be divided into the 
two forces, AC = P cos. a, and DC = Psin.a«; the force, BG, likewise into 
BE = Wcos.f, and EG= Wsin. &. The proportion then becomes P 
cos.a:Weos.8::BF:FA. This. proportion only holds good, however, 
when the lever can only turn on the fulcrum without shifting. Should it 
lie but loosely upon the fulerum, there must be equilibrium of the horizontal 
part of the forces, and the proportion P: W::sin. 8: sin. a. 
Among the numerous applications of levers of the first class is to be 
reckoned the balance, that arrangement by which the weight of a body is 
determined. The common balance consists of an equal-armed lever, in 
which the two forces—the body, P, to be weighed, and the weight W— 
must act perpendicularly to the two arms of the lever. In the equality of 
the arms of the lever, the forces must necessarily be equal, that is, the 
weight to the weighed. When one beam of the balance is longer or 
heavier than the other, by even a very slight amount, the equality of the 
weight and the object weighed is destroyed, and the balance is false. 
In the steel yard (pl. 16, fig. 24) other conditions of equilibrium exist. 
In this the beam, AB, is a lever of unequal arms; the arms, BF and AF, 
are supported at F’, where the balance is either suspended as in the figure, 
or else held in the hand.. A definite proportion exists between the lengths 
of the two arms, as 1:4 or 1:10, &c., and the forces will, according to the 
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