210 
MR. L. X. G. FILOX OX THE ELASTIC EQUILIBRIUM OF 
cylinder are in general made up of three sheets, and they fall into two classes : 
(o) those for which ZZ has a value less than a certain critical value, which, as nearly 
as I can find out from graphical methods, is about I'Ol Q, and (6) those for which ZZ 
has a value greater than UOl Q. 
The surfaces (a) consist of two solid caps or buttons, round the centres of the end 
sections, together witli a hollow cylindrical shell surrounding the middle of the 
cylinder. For values sensibly < ‘9 Q the latter sheet disappears, and only the caps 
remain, tlieir volume gradually dwindling down to zero as ZZ falls to ‘686 Q. 
The surfaces (h) consist of an elongated core, resembling a cylinder closed by 
curved ends, surrounding tlie centre of the compressed block, together with two 
annuli at the ends, as shown in the figure i-eferred to. 
The critical surface ZZ I'OlQ consists of two nearly plane sheets, roughly co¬ 
inciding with the cross-sections Z = i 5 c/6, and one cylindrical sheet, which bends 
inwards towards the end, though without completely closing in, and which roughly 
coincides with the cylinder r = 2o '3 over the greater part of its surface. 
§ 23. Aj^pUeativn to Ixvpture, Distribution of Maximum Stress, Strain, and 
Stress Difference. 
In consideiing >vhat ha])pens A\dien a material breaks, Ave haA’e to ask, first of all, 
Avhether it be brittle or ductile. In the first case, the laAv of stress to strain will be 
approximately linear uja to the point Avhere rupture takes place ; in the second case, 
the stress-strain relation remains approximately linear until a point is reached (called 
the yield-])oint) at Avhich a large and sudden change occurs in the stress-strain curve, 
after which the material l)ecomes sensil)ly ])lastic, so that rupture finally takes place 
after a large permanent deformation. 
In a])})lying an elastic tlieory to practiee, Ave can, in strictness, treat of rupture 
only in the case of a Inlttle solid. Even then it lias t(A be borne in mind that the 
mathematical theory of strains—upon Avhich the equations of elasticity depend— 
re(:[uires the strains to be so small that their squares are negligible. It is possible 
that, eA’en in the case of the most lirittle solids knoAAn, this condition may cease to 
hold before rupture occurs, although the stress-strain relation may continue to be 
linear. NeA'ertheless, the calculated A'alues of eA-en the lireaking strains in a material 
like cast iron, for instance, are so small as to render this unlikely. 
For a ductile metal, such as mild steel, the elastic results only tell you Avhere the 
material Avill begin to take permanent set. 
In the case of stone or cement, hoAveA^er, to AA'hich the present results AA'ould be 
applied, there seems to le no definite yield-point or elastic limit, the material being, 
in fact, only imperfectly elastic throughout. Still, aac may consider that the results 
