MECHANICS. 21 
sented in fig. 8, will be = LW cos.«; for that in fig. 11 =4 LW see. 22, 
and for that in fig. 12, =} LW sec.? a. 
The preceding figs have had reference to the conditions of equili- 
brium of beams supported at both ends and loaded in the centre; we will, 
now consider the case where the load is applied elsewhere than in the mid- 
dle, as in pl. 16, fig. 50. The weight appended may then be supposed to 
be divided into two weights, which act on the arms of levers whose lengths 
are as the parts of the beam. Thus, representing by L the entire 
mn 
1 re we W = 
length of the beam, m, and n its parts, then the pressure sais 
L Supposing two equal or different weights applied at different points, 
as in fig. 51, and calling the distance from the left point of support to the 
left point of suspension of the weight, m; that from the left point of suspen- 
sion to the right point of support, x; that from the left point of support to 
the right point of suspension, r; and that from the right point of suspension 
to the right point of support, 0; then for the first weight the pressure will 
be F a and for the second F = _ , where W and W’ are the 
corresponding weights, and L the length between the points of support. 
To obtain the pressure resulting from this double pressure, upon every other 
point of the beam, call the distance of this point from the left point of sup- 
port, s, and that from the right, ¢, and we will have the following pro- 
portion: n:t: ii Bie plas for the pressure exerted by the left weight ; 
and o:s: eee pees for that of the right ; hence the combined pressure 
einthiin shied) ping a . 
An application of this proposition is to be found in fig. 49, where the 
weight acts upon the middle of an inflexible bracket. Here the effect of 
this weight upon the beam is the same as if two weights of half the origina! 
one were suspended at the points where the bracket meets the beam. It 
will be easy, from the preceding, to determine the value of 1" in the middle 
of the beam, where, as in pl. 16, fig. 52, several equal weights are suspended. 
It also follows, that when the burden is distributed uniformly over the 
whole beam, its action is the same as if half the amount were attached to 
the centre of the beam. 
The beams hitherto considered have been, for the most part, such as 
were supported at the ends; and we have found that such a beam is four 
times as strong as the same beam attached to a wall by one extremity and 
loaded at the other. Supposing the beam to be walled in at both ends, as 
in pl. 17, fig. 12, and loaded by the weight Q, we may assume that it will 
break at the same instant in A, B, and C, provided Q be of sufficient 
amount. Represent the forces which produce fracture at these three points 
by p p’, p’. and the two parts of the beam by a, a’, the total length of the 
198 
