22 PHYSICS. 
beam by L, its breadth by 6, and its depth by hk. Fracture will then ensue 
: bh? , bh 
when, according to the preceding formula, p =} m gel t= Baad and 
; bLh? 
Regi f aca Q, however, must be sufficient to produce all three frae- 
| f la L 
° a PS xD 722 ors 4 - a8 2 : 
tures ; therefore, Q —1mbh & 1 choke | ; or, as a’ = L—a,Q =—2m 
2 
wa . Calling the distance by which the point, C, lies out of the centre, 
d, then will Q —2 ivi ; 1f d=0, or if C lie in the middle, then 
Q =m 7 Hence it follows from this formula that beams loaded in the 
middle are weakest, but that they can support eight times as much as when 
attached at one end and loaded at the other. 
For the case in which the beam, as in fig. 13, is inclined at an angle, as 
BAD =a, to the horizon, the perpendicular lateral force, CG = Q cos.a, 
can alone tend to produce fracture; the other lateral force, CF = Q sin. a, 
involving the strength of crushing: Q becomes then = 1m — 
Those bodies which in all their sections present the same strength are of 
great importance: the bodies of equal resistance. The fracture of bodies 
of equal section throughout occurs always at the surface of attachment, or 
where the weight is attached; consequently the transverse sections lying at 
a distance from these points are too great, and must be diminished. Such 
a case has been considered (fig. 5) under the head of absolute strength; it 
remains here to mention seme others._ Fig. 14, pl. 17, represents a body 
which, fixed at one end, is loaded at the other with the weight Q, and where 
transverse sections are throughout, rectangles of equal breadth: represent- 
ing the height by y, the breadth by z, and the distance from C of the section 
MN by z, then, according to the preceding nomenclature, AB =h, AC =L, 
and z — 6: we then have am =~, hence y? = as This, however, is 
the equation of the parabola; and the outline, BC, must be a parabola, 
2 
h? 
whose vertex lies at C, and whose parameter ie Pl. 17, fig. 15, repre- 
sents a similar body, ABC, upon which the weight, Q, is uniformly distri- 
buted. Here the same references are employed, and we have for y in the 
; ‘ h ' 
section MN, the value y = *" .x, whence it follows that the outline, BC, 
4 
must be a straight line. Finally, suppose fig. 16 to represent the body, 
AB, resting freely at its two extremities, its sections rectangles of equal 
breadth, and the weight, Q, moving longitudinally above the body ; required 
the conditions according to which the inferior curve line is formed. Let 
AC = BC =} L=a, CD =f, and for any given section, MN, CM =z, 
2 
and MN —y; then 7? == (a* —2x?), and the curve of outline will be a 
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