MECHANICS. 19 
g equal to the distance from the axis of the most extended fibre, 6, then 
will ST 1918 .u, and the force g, producing this extension, will —A* : Us 
& 
here A is the absolute strength, and E the modulus of elasticity, or the 
weight necessary to stretch the body to double its length. GF is, however, 
composed of an innumerable number of fibres, whose sum, FH, may be 
represented by h’, and the force, P, necessary to extend all these fibres will 
E £6 bh’? 
The compressing force, P’, for the part below the axis, 
Se 
whose modulus of elasticity, or force required to —— it to half its 
h—h’')? 
original length, may be represented by EB’, will be = ies me = . The 
13 ! 
statical moments of the two forces are, Py = lice ee and P’y = 4 z 
"eg 3 a § 
b. agile The statical moments, however, of the weight Q, whose 
leverage, FL = z, will then necessarily be Qz ear 3 (Eh 3 4+ E’(h—h’)3). 
Since the fibres at F experience no compression, = fi P’ will = O, or Eh’2 
= E’ (h—h’)2, Qz then becoming = is ae ze ud 
Producing GG’ and HH’, until they intersect at U, then UF will be eh 
radius of curvature, e, for the arc element, EF =A, and al aa an nd ; 
= = this value substituted in the formula for Qz, and oh taken for h’, 
where 9 is a magnitude dependent upon the situation of the neutral axis, 
and expressing the ratio of extensibility and compressibility, we will have 
a ee a The right side of this equation is constant for equal 
parallelopipeda, and depends upon the elasticity of the body ; it is called the 
moment of elasticity = W. Let Q be the mean of several forces, then Qz, 
the sum of their moments, will = M, and Me=W;; that is, for every 
transverse section at right angles to a bent parallelopipedon, the product 
of the radius of curvature by the moment of the force, is a constant 
quantity. 
In most cases, however, the bending of the body is so slight, that the 
leverage, x, of the weight Q, may be exchanged for the length, FA = 1, and 
’ 
c 
tit we thus obtain, by introducing this quantity into one of the pre- 
m 
ceding equations, Ql = aoe W. Suppose now the body (pl. 17, fig. 6) to 
& 
be fixed in the plane HH’, the preceding formule will give the moment of 
the weight, Q, which can break off the body, HDD’H’, at the plane HFH’; 
Q is also the relative or transverse strength of the parallelopip:don. The 
co-efficient of fracture, m, must be obtained by trial. Assumi' -; the neutral 
ICONOGRAPHIC ENCYCLOPZDIA.—VOL, I. 13 193 
