Physics and Mathematics to Geology. 241 



suppose that the maximum stress-difference aiid greatest strain 

 are to be compared not with the values that answer to rup- 

 ture, but either with those that answer to the limit of perfect 

 elasticity or with those derived by dividing the values an- 

 swering to rupture by some factor of safety. This factor 

 must then be large enough to prevent the limit of perfect 

 elasticity being passed. Thus from either point of view we 

 encounter a formidable difficulty, viz. the uncertainty of what 

 is the limit of perfect elasticity. 



We have supposed that a bar may be brought into a state 

 in which it is perfectly elastic for longitudinal tractions not 

 exceeding L 2 . Answering to this we have L 2 for the stress- 

 differance, and L 2 /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 2 is the 

 limiting value of the stress-difference, the other that L 2 /E is 

 the limiting value of the greatest strain for all possible stress- 

 systems in materia] of the same kind as that in the bar. The 

 complete experimental 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 w T ould 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 determining the range of the 

 mathematical theory. It will certainly not exceed the phy- 

 sical 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 practical 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 safeguarded 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 mathe- 



