PHYSICS AND MATHEMATICS TO GEOLOGY. 13? 
fulfilled, so long as the stress does no exceed a certain limit. Again, 
the fact that a large mass of metal is sensibly isotropic is no sufficient 
reason for attributing isotrophy to the same metal when rolled into 
thin plates or drawn into thin wires. 
It is quite possible that the three conditions (A), )B), and (D) repre- 
sent an ideal state which is never actually reached, and that a diver- 
gence may always be shown by the use of very delicate apparatus. If 
this be true, then the results obtained by the mathematical theory can 
not claim absolute correctness. It seems however to be satisfactorily 
established that many materials in the state of ease satisfy these con- 
ditions with at least a very close approach to exactness, so that the 
results of the mathematical theory when properly restricted are then 
sufficiently exact for practical purposes. 
From the preceding statements it will be seen that it is of the utmost 
importance to know what are the limits within which the conditions 
assumed by the mathematical theory are satisfied with sufficient 
exactness to justify its application. This question must of course be 
settled by experiment, but it is beset by various difficulties which ought 
to be clearly recognized. These arise in part from the serious obstacles 
in the way of a complete experimental knowledge, and in part from 
the want of a proper understanding between those interested in the 
practical and theoretical sides of the subject, and a consequent confu- 
sion in the terms used. . 
To avoid complication let us begin by supposing the mathematical 
limit of perfect elasticity to coincide with the physical. Let us con- 
sider the simple case of a bar under uniform longitudinal traction. We 
may suppose the bar isotropic, and in consequence of suitable treat- 
ment perfectly elastic for loads not exceeding L; No mechanical 
treatment, we Shall suppose, can render it perfectly elastic for loads 
ereater than Ly. It does not follow that a load L, will necessarily rup- 
ture the bar either immediately or in course of time, but simply that 
for any load greater than L, the strain is not perfectly elastic. Increas- 
ing the load from zero we should reach a load L;, probably greater 
than L,, that would in process of time rupture the bar, or a load Ly 
greater than L; that produces immediate rupture. All these loads are 
supposed to refer to unit of area. 
Now in the initial state of the bar we should be entitled to apply the 
mathematical theory only until the load L; was reached. When we 
aim at finding the utmost capability of the material under longitudinal 
load, we may perhaps apply the theory until the load L, is reached, but 
here we must stop. ‘To apply it until the loads L; or L, are reached— 
assuming these greater than L,—is clearly inadmissible. 
Results of a similar kind hold for all the comparatively simple forms 
of stress—such as pure compression, torsion, or bending—in which 
practical men are interested. There are limits to the state of perfect 
elasticity lower than the limits at which rupture takes place, at least 
immediately. 
