EQUILIBRIUM OF ELASTIC SOLIDS. 119 
and since (g—p), R and S are known, and since at the surface, where @, (zx, y)=0, 
p=0, all the data are given for determining the absolute value of p by integration. 
Though this is the best method of finding p and qg by graphic construction, it 
is much better, when the equations of the curves have been found, that is, when 
g, and ¢, are known, to resolve the pressures in the direction of the axes. 
| The new quantities are p,, p,, and g,; and the equations are 

tand=—%_,  (p—gP=9,2+(p,—p,), Py + Pn =P +9 
Pi-P2 
It is therefore possible to find the pressures from the curves of equal tint and 
equal inclination, in any case in which it may be required. In the meantime 
the curves of figs. 2, 3, 4 shew the correctness of Sir Joun HERScHELL’s ingenious 
explanation of the phenomena of heated and unannealed glass. 






















Nore A. 
As the mathematical laws of compressions and pressures have been very thoroughly investi- 
gated, and as they are demonstrated with great elegance in the very complete and elaborate memoir 
of MM. Lame and Crarryron, I shall state as briefly as-possible their results. 
Let a solid be subjected to compressions or pressures of any kind, then, if through any point in 
the solid lines be drawn whose lengths, measured from the given point, are proportional to the com- 
pression or pressure at the point resolved in the directions in which the lines are drawn, the extre- 
mities of such lines will be in the surface of an ellipsoid, whose centre is the given point. 
The properties of the system of compressions or pressures may be deduced from those of the 
ellipsoid. 
There are three diameters having perpendicular ordinates, which are called the principal ames 
of the ellipsoid. 
Similarly, there are always three directions in the compressed particle in which there is no tan- 
gential action, or tendency of the parts to slide on one another. These directions are called the 
principal axes of compression or of pressure, and in homogeneous solids they always coincide with 
each other. 
The compression or pressure in any other direction is equal to the sum of the products of the 
_ compressions or pressures in the principal axes multiplied into the squares of the cosines of the 
angles which they respectively make with that direction. 
Nore B. 
The fundamental equations of this paper differ from those of Navier, Poisson, &c., only in not 
‘assuming an invariable ratio between the linear and the cubical elasticity; but since I have not 
attempted to deduce them from the laws of molecular action, some other reasons must be given for 
adopting them. 
The experiments from which the laws are deduced are— 
1st, Elastic solids put into motion vibrate isochronously, so that the sound does not vary with 
the amplitude of the vibrations. 
2d, ReGNauiy’s experiments on hollow spheres shew that both linear and cubic compressions 
__ are proportional to the pressures. 
3d, Experiments on the elongation of rods and tubes immersed in water, prove that the elon- 
_ gation, the decrease of diameter, and the increase of volume, are proportional to the tension. 
VOL. XX. PART I. 21 
