
’ EQUILIBRIUM OF ELASTIC SOLIDS. 98 
On the face z, —g,dedy aig —p,dudy 
dq, dp 
%| (m+ Gidadady | (g' ts P dadzdy (pn + Pd2)dady 
Attractions, pXdadydz pYdadydz pZdadydz 
Taking the moments of these forces round the axes of the particle, we find 
W=h W=h %=%3 
and then equating the forces in the directions of the three axes, and dividing by 
dx, dy, dz, we find the equations of pressures. 
dp, , 4d , 4% 
dpe dg. ae. 
dq, 44, 
dz dx 
i (3.) 
Fi + G+ G+ pu=o0 
The resistance which the solid opposes to these pressures is called Elasticity, 
and is of two kinds, for it opposes either change of volume or change of figure. 
These two kinds of elasticity have no necessary connection, for they are possessed 
in very different ratios by different substances. Thus jelly has a cubical elasticity 
little different from that of water, and a linear elasticity as small as we please ; 
while cork, whose cubical elasticity is very small, has a much greater linear 
elasticity than jelly. 
Hooxe discovered that the elastic forces are proportional to the changes that 
excite them, or, as he expressed it, “ Ut tensio sic vis.” 
To fix our ideas, let us suppose the compressed body to be a parallelopiped, 
and let pressures P,, P., P$ act on its faces in the direction of the axes a, £, y, 
which will become the principal axes of compression, and the compressions will 
he Oa 6B by 
+ pX=0 
Equations of Pressures. 

+ 
+ pY 
The fundamental assumption from which the following equations are deduced 
is an extension of Hooxe’s law, and consists of two parts. 
I. The sum of the compressions is proportional to the sum of the pressures. 
II. The difference of the compressions is proportional to the difference of the 
pressures. 
These laws are aes by the following equations :— 
L @,+P,+P,)= =3n (22. a8 4ST) (4) 
Equations of Elasticity. 
[@,-Py =m (52-38 
II. @, —P) =m (B - 57) coal 
em en(t 2) 
