134 “SOME APPLICATIONS OF 
We have supposed that a bar may be brought into a state in which ~ 
it is perfectly elastic for longitudinal tractions not exceeding L,. An- 
swering to this we have L, for the stress difference, and L,/E for the 
greatest strain. Now if the two theories described above really apply 
to the limit of perfect elasticity, the one would seem to maintain that 
L, is the limiting value of the stress difference, the other that L./K is 
the limiting value of the greatest strain for all possible stress systems 
in material of the same kind as that in the bar. The complete experi- 
mental proof or disproof of such theories is not likely to be easy. Thus 
taking for instance the case of longitudinal traction, suppose it were. 
shown that a certain method of treatment which raises the elastic limit 
for load parallel to the axis of a bar does not raise the elastic limit for 
longitudinal load in a bar whose length lay in the cross section of the 
original bar. This would only suffice to prove that the treatment 
adopted did not give a fixed elastic limit the same for all kinds of 
strain, it would leave the possibility of such a limit being obtained in 
some other way an open question. 
In the preceding remarks the mathematical and physical limits of 
perfect elasticity have been supposed identical. When they differ, the 
mathematical limit is of course that which must be employed in deter- 
mining therange of the mathematical theory. It will certainly not ex- 
ceed the physical limit. I may add that while for certain structures 
such as isolated boilers the physical limit may most nearly concern the 
practical engineer, in other structures, such as girder bridges, the 
stress-strain relation is assumed to be linear in designing the several 
parts, so that the first mathematical limit is then of the utmost practi- 
eal importance, 
In the previous discussion of the stress-difference and greatest-strain 
theories, as settling the limits of application of the mathematical 
theory, it has been taken for granted that the condition (C) was safe- 
guarded by them. Now in most ordinary systems of loading this is 
probably the case, but it is not always so. For instance, if we assume 
the mathematical theory to hold, a solid isotropic sphere under a 
uniform surface-pressure shows none but negative strains, and the 
three principal stresses are everywhere equal. Thus the greatest 
Strain is everywhere negative, and the stress difference everywhere 
zero. This is true irrespective of the magnitude of the surface pressure, 
and so, according to both theories, the stress-strain relation would be 
linear and the mathematical theory would apply, however large the 
pressure was. According to the theories, one might continue to employ 
mathematical formule which indicated a reduction of the sphere to one- 
millionth of its original volume. It is obvious, however, that a re- 
duction of the volume by even a tenth would produce strains which aie 
probably far in excess of those admitted by the principle (C). In 
formulating an objection to the universal application of the theories, I 
have preferred to attack them on the side of the principle (C) so as to 
