MECHANICS. 25 
any whole number, it follows, if L represent the length of the rod—z, 
that L= ien/ 0, and fob, Laws/% and Q = . As, how- 
ever, Q is independent of the amount of the bending, this weight, in any 
degree of bending, holds the elasticity of the body in equilibrium, or Q is the 
capacity of cracking of the rod. 
Combining these values with those previously obtained by substituting the 
moment of elasticity for W, we find that in prismatic beams of homogeneous 
material, the capacities of cracking are as the breadths, as the third power 
of the thicknesses (least sides), and inversely as the squares of the lengths; 
in cylinders, as the fourth powers of the radii, and inversely as the squares 
of the lengths. 
With respect to the strength of torsion, or twisting, let us suppose a body 
( fig. 21, pl. 17) fixed at one of its ends, AA’, and a force, P, acting at the 
other extremity on the arm of a lever, CD=—R, capable of producing a 
rotation about the axis, CC. If, now, the diameter BB be twisted to B’B, 
AA’ will be stationary ; the homologous diameters, however, of all interme- 
diate sections will be displaced in proportion to their distance from the 
surface of attachment. The angle BCB’ is then the angle of rotation, and 
the turning force must be strong in proportion to the amount of this angle, 
to the strength of the transverse section of the fibres, and to the distance of 
the fibres from the axis of rotation; the longer the fibres, however, the less 
need be the force. 
An actual twisting apart of the body must ensue when the remote fibres 
can yield no more without being actually ruptured; and in cylinders of 
homogeneous material, the statical moments of the forces which produce 
such a rupture by twisting, are as the cubes of the radii. 
B. Dynamics or Sour Boptss. 
The theory of motion is much more difficult as well as more compre- 
hensive than that of equilibrium: it calls mathematics into play to a much 
greater extent, and this in its most abstruse branches. 
The motion of a body, which may result from one or several forces, is, in 
respect to its direction, either rectilineal or curvilineal; in respect to its 
velocity, either uniform or variable. Motion is said to be equable when 
equal spaces are traversed in equal times: when, for example, the same 
amount of space is passed over in each successive second. Of this kind is all 
motion produced by a single force acting instantaneously—in a blow, for 
instance—provided that the motion meet no obstruction. Motion is variable 
when, instead of remaining the same, it increases or diminishes. If the 
motion increase or diminish equally in equal times, it is said to be uniformly 
accelerated or retarded. 
Tne force itself producing motion may be either momentary or con- 
tinuous. In the former the force is to be considered as acting for a very 
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