220 
MR, G. H. DARWIN ON THE STRESSES 
power of gravitation and to be superficially corrugated. In consequence of mathe¬ 
matical difficulties the problem is here only solved for the particular class of surface 
inequalities called zonal harmonics, the nature of which will be explained below. 
Before discussing the state of stress produced by these inequalities, it will be con¬ 
venient to explain the proper mode of estimating the strength of an elastic solid under 
stress. 
At any point in the interior of a stressed elastic solid there are three lines mutually 
at right angles, which are called the principal stress-axes. Inside the solid at the 
point in question imagine a small plane (say a square centimeter or inch) drawn 
perpendicular to one of the stress-axes; such a small plane will be called an inter¬ 
face. " ,r The matter on one side of the ideal inter-face might be removed without 
disturbing the equilibrium of the elastic solid, provided some proper force be applied 
to the inter-face; in other words, the matter on one side of an .inter-face exerts a force 
on the matter on the other side. Now a stress-axis has the property that this force 
is parallel to the stress-axis to which the inter-face is perpendicular. Thus along a 
stress-axis the internal force is either purely a traction or purely a pressure. Treating 
pressures as negative tractions, we may say that at any point of a stressed elastic 
solid, there are three mutually perpendicular directions along which the stresses are 
purely tractional. The traction which must be applied to an inter-face of a square 
centimeter in area, in order to maintain equilibrium when the matter on one side of 
the inter-face is removed, is called a principal stress, and is of course to be measured 
by grammes weight per square centimeter. 
If the three stresses be equal and negative, the matter at the point in question is 
simply squeezed by hydrostatic pressure, and it is not likely that in a homogeneous 
solid any simple hydrostatic pressure, absolutely equal in all directions, would ever 
rupture the solid. The effect of the equality of the three stresses when they are 
positive and tractional is obscure, but at least physicists do not in general suppose 
that this is the cause of rupture when a solid breaks. 
If the three principal stresses be unequal, one must of course be greatest and one 
least, and there is reason to suppose that tendency of the solid to rupture is to be 
measured by the difference between these principal stresses. 
In one very simple case we know that this is so, for if we imagine a square bar, of 
which the section is a square centimeter, to be submitted to simple longitudinal ten¬ 
sion, then two of the principal stresses are zero (namely, the stresses perpendicular to 
the faces of the rod), and the third is equal to the longitudinal traction. The traction 
under which the rod breaks is a measure of its strength, and this is equal to the 
difference of principal stresses. 
If at the same time the rod were subjected to great hydrostatic pressure, the break¬ 
ing load would be very little, if at all affected; now the hydrostatic pressure subtracts 
* This term is cine to Professor James Thomson. 
