er ae ae PI Pn A 
and Strength of Materials. 239 
_ Let ACBD (figure 4th) be the section of 
fracture, whose breadth is 6, and depth d, and 
let a denote the distance of the neutral line 
AB from the top of the beam ; then d—a will 
be its distance from the bottom. Call any 
intermediate distance from AB, onthe line of 
extension, as that of de, x; and any. dis- 
tance on the line of compression, as that of 
x, y. Then (if we put s= the weight sus- 
tained by a fibre at the top, and r =the co- 
temporary resistance of one at the bottom, 
and v and w = the powers of extension and 
compression, which are as their forces respec- 
tively,) we shall have, as before, a: 2° :: 
Ce og 
3% = force exerted by a fibre at distance 
a’ 
line, and -may be_very plainly seen if we estimate by this 
rule, the strengths of bodies that suffer a slight compres- 
sion, and. compare the results with the known strengths of 
the same bodies, if they had been wholly incompressible. 
For example :—The strength of an incompressible joist, 
2 
broke by a weight hung at one end, being io (arti- 
cles 4th and 9th,) where sis the longitudinal strength of the 
fibres in a unity of surface, 6 the breadth of the piece, a 
its depth, Zits length, and wv the index of extension: the 
ah eo a tine hel hg one according to Mr. Barlow 
will be Te eh where d is the depth of the area of tension, 
and the rest as before. 
