PHYSICS AND MATHEMATICS TO GEOLOGY. 133 
point—a pressure being reckoned negative—attains a certain value, 
rupture will ensue at this point. Thus, if in descending order of mag- 
nitude, the principal stresses at a point be T,, T., T;, then T;—T; is the 
stress difference* at this point, and the theory asserts that rupture will 
ultimately ensueif the stress difference anywhere equals L;, the load for 
ultimate rupture of a bar of the material by longitudinal traction; while 
if the stress difference anywhere equals L,, the load for immediate rup- 
ture by longitudinal traction, then immediate rupture will ensue. 
The second theory, which is supported by the great authority of de 
Saint-Venant,t replaces the stress difference of the first theory by the 
ereatest strain. It thus asserts that the condition for rupture is found 
by equating the largest value found anywhere for the greatest strain 
to the longitudinal strain answering to longitudinal traction L;, or to 
that answering to the traction L,, according as the rupture is ultimate 
or immediate. This theory maintains that extension in some direction 
is necessary for rupture. 
The two theories may, as in the case of pure longitudinal traction, 
lead to the same result; but in general they do not, so one at least of 
them must be wrong. When we examine the theories, still supposing 
the mathematical and physical limits of perfect elasticity the same, a 
very obvious difficulty presents itself. It is assumed that the stress- 
difference and greatest strain are derived by the mathematical theory; 
but that theory applies only so long as the material is everywhere per- 
fectly elastic, whereas rupture, at least when immediate, presents itself 
after the elastic limit has been passed. Thus if the application of the 
mathematical theory leads to values for the maximum stress difference 
and greatest strain equal to the values of these quantities answering 
to rupture, at all events when immediate, the true conclusion would 
seem to be that the fundamental hypothesis on which the treatment 
proceeds, viz, that the material follows the laws assumed by the mathe- 
matical theory, has been shown to be incorrect. Nothing has been 
proved except that the elastic limit must be passed and that the mathe- 
matical theory does not apply. 
The only logical way of interpreting the theories is to suppose that the 
maximum stress difference and greatest strain are to be compared not 
with the values that answer to rupture, but either with those that 
answer to the limit of perfect elasticity or with those derived by dividing 
the values answering to rupture by some factor of safety. This factor 
must then be large enough to preventthe limit of perfect elasticity being 
passed. Thus from either point of view we encounter a formidable diffi- 
culty, viz, the uncertainty of what is the limit of perfect elasticity. 
* See Prof. Darwin, Phil. Trans., 1882, pp. 220, 221, ete.; also Thomson and Tait’s 
Nat. Phil., vol. 1, part 1, p. 423. 
t See Pearson’s The Elastical Researches of Barré de Saint-Venant, art. 5 (c¢), ete. 
t Ibid. art. 4 (v), 5 (a), &e. 
