BEAM OF RECTANf4TTLAB CROSS->SECTION UNDER ANY SYSTEM OF LOAD. 141 
y' — ^ 2 } is everywhere continuous. Its differential coefficient with regard 
to y is indeterminate at (ffi a!, 0); if we proceed along y' = 0 it increases hy tt as we 
pass the point (— a', 0) and decreases hy tt as we pass tlie point (+ a', 0). The 
same holds with regard to {x + - {x - a') ^ and its differential coefficient 
with regard to x. 
Hence, as far as U and are concerned, tliey are lioth finite, continuous, and one¬ 
valued tliroughout the beam. are everywhere finite, Imt are indeterminate 
at {± o/, 0). As we proceed along ?/' = 0, decreases abruptlv liy ^ - -p — 
dy ^ •' 2 \ Y+ ^ ^ 
as we pass (— o/, 0) and iiicreases again by tlie same amount as we pass (+ 0). 
Similarly ^ decreases Iw -- 
dx ^ 2 X' + /i 
as we pass (— a\ 0), and increases hy the same 
amount as we pass (+ a', 0). The first of these results means an alnaipt change in 
the angle at which the distorted cross-sections meet the horizontal, and tlie second 
shows that the distorted form of the upper edge of the beam receives a sudden 
inflection downwards as v^e enter the layer of shear, and is again suddenly inflected 
upwards as we emerge from it. 
It has lieen shovm in a paper hy the author “ On the Equilibrium of Circular 
Cylinders under Certain Practical Systems of Load” Phil. Trans.,’ A, vol. 198, 
pp. 14/-233), that a jirecisely similar occurrence takes place in a circular cylinder 
subjected to a uniform ring of shear, over a certain length of its curved surface. The 
law that shear depresses the parts of the surface towards which it acts appears to 
be a general one. 
Passing on now' to consider the stresses P, Q, S, we find that Q and S remain 
everywhere finite, liut are indeterminate at the points (ffi a', 0). If ive keep to 
2 /'= 0, Q is continuously zero and S changes hy L at (ffi a', 0),. as it should. 
But P not only contains a part which becomes indeterminate at (ffi a'^ 0), it also 
2L r 
contains a term — ~ log -- wliich becomes Infinite at those points. 
This is a result for ■which we had no analogue in the case of a uniform layer of 
pressure. In that problem the stresses were everywhere finite. We now see that any 
finite discontinuity in the shear introduces an Infinite pressure or tension P in tlie 
neighbourhood of this discontinuity. This result, again, has been found to hold 
good for circular cylinders. It may he laid down as an absolute rule that for an 
engineering structure to be safe, there should never occur any discontinuity in the 
shearing stress across any surface inside the material or on its boundary. It is true 
that in most cases the stress will he relieved liy plastic flow' and the variation of the 
shear will become continuous, though rapid. But such points, especially tlie point 
from ivhich the shear starts acting (— o', 0), where tlie infinite stress is a tension. 
■will remain points of weakness and danger, 
