24 - PHYSICS. 
= (¢ L°—4 2°) z, and u= ee » if Q again = 0, then will y = oa (Li 
50 Lt : 
6 L z ae 4 = op " A = i, ‘ S 2 
2L2’4..c’) and u eae ssuming Q = pL, then the depression in the 
two cuses will be as 8:5; consequently, when a weight is distributed 
uniformly along the whole fibre, the depression will be only 3 of what would 
result from the application of the same weight to the middle. 
In investigating the strength of resistance to a crushing force, we suppose 
prismatic bodies standing vertically, upon whose upper extremities weights 
are laid, and then investigate the force necessary for crushing, and that 
which produces first a bending, and then a cracking. With respect to the 
force of crushing, it appears, from experiment, to increase in a somewhat 
greater ratio than the cross section, although it may be properly assumed 
that if all parts of the cross section experience equal pressure, the force will 
be proportional to the cross section. Calling, therefore, the strength 
(obtained by trial) of a certain cross section, m, and the area of the prism 
to be investigated, A, then Q—mA. The capacity for being crushed 
diminishes as the circumference increases, the area remaining the same ; it 
is, therefore. least in the circle: it is less, also, as the form of the body 
_approaches in height to the cube. 
To obtain the law of cracking, let us suppose an elastic rod, AB (pl. 17, 
fig. 19), which, fastened at A, assumes naturally the vertical direction AZ ; 
becoming bent, however, into the curve ADB by a weight attached to the 
upperend,B. To find the co-ordinate equation of this curve, assume the verti. 
cal direction, BC, of the weight as the axis of abscissas, and B as their origin. 
For any point, D, of the curve whose radius of curvature is e; let BQ = 2, 
DQ = y, and AC = a, and let the curvature of the rod be so slight that the 
abscissa may be exchanged for the length of the arc. If, now, y be the 
leverage of Q, then M = Qy, and Qy == iis By assuming another point of 
§ 
the curve, F’, infinitely near to D, and bringing into the calculation the quanti- 
ties FH, DH, with their trigonometrical proportions, we finally obtain for x 
Q 
the value wie are. (sin. = Swe ) where the one factor is an arc 
whose sine is ey to the quotient of the two radical quantities, o indicating 
the angle at which the geometrical tangent of the point A meets the curve. 
v Wt a, 
For y we have the value ace ae tem yu, hen Most generally a 
is to be taken = 0, or the direction of the bending weight passes, as in 
pl. 17, fig. 20, through the point of attachment, A. The equation then becomes 
oa are. (sin. SY a and yar sin. (« Ve ). 
tzo 
For the points A and B, y = 0, thus x= We are (sin. = 0); as, how- 
ever, arc (sin. = 0) may be taken —0, 1, 27, 38% — - — iv, where 7 represents 
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