212 
MK. L. X. G. FiLOX OX THi: ELASTIC EQUILIBRIUM OF 
so that the greatest s will correspond to the greatest T, if Tn Tg have the same 
sign. This is our case everywhere, except in cases where (f)(f) > 0, and then ^<j6 is so 
small that it still leaves the strain corresponding to ZZ numerically the greatest. 
We have then, remembering we have assumed X = p,, to investigate the values of 
ZZ — ( ZZ lITv T" ^ 4 *) ~ 2p.s'.. 
This will he proportional to the greatest strain, except near 5: = ± = 0, where 
ini — 1 (ZZ -|“ lilt -j- 4 ^ 4 *) ~ 
should he taken. It is found, however, that at this point the strain is comparatively 
small, and the maximum strain there is a matter of inditference. 
'Fable of s~/s, where s = maximum stretch under the same uniform pressure. 
= 0. 
,r = c/Q. 
~ = 2c/6. 
= 3c/6. 
.? = 4c/6. 
.r = 5f/6. 
r = c. 
0 
1-13245 
1-12729 
1-10755 
1-05938 
■95338 
•77296 
■23743 
rt/3 
1-10000 
1-09589 
1-07935 
1-03756 
•94751 
•76590 
• 39034 
■2^13 
1-01133 
1-01035 
1-00509 
•98813 
•94446 
•86416 
•78311 
a 
-9024G 
•89563 
•87724 
•85716 
•86591 
•99395 
1-57685 
It we take therefore the “ greatest stretch 
at the perimeter of the ends, hut this time only when the stress is 
tlieory, failure of elasticity still occurs 
1 
1-577 
(limiting- 
stress in the case of uniformly compressed cylinder), so that although the apparent 
strength is less than in tlie uniform case, it is greater than if we adopt the “ greatest 
stress ” theory. 
'Fhe lines of ecpial })rincipal stretch s./s ai-e shown in Diagram 13. They are 
drawn for only one quarter of the meridian plane, the rest being symmetrical. They 
present the same general characteristics as the curves of equal stress ZZ, with this 
difference, that the critical line corresponds to S; = -915 s. Again, the caps or buttons 
at the ends are far larger; so that if pieces are cut out, they will be considerably 
larger than on the “greatest stress” theory. Also, looking at the inclination of the 
lines joining the corners to the critical points, i.e., the points where the two branches 
of the critical line intersect, ^ve see that the fragments, if approximately conical near 
their base, will probably l)e cut otf at a mucli higher angle than in the previous case. 
Let us now jiroceed to consider what happens if we adopt the third or “ greatest 
stress-difference ” theory of rupture. It is easy to see from the tables of RE, ZZ, 
and 4>4^ that the greatest stress-difference Is either ER-ZZ or 4^4^-ZZ. In the 
sixteen cases tabulated, for which a. > c/3, the first of these is the greatest .stress- 
