THE EQUILIBRIUM OP ELASTIC SOLIDS. 71 



There are three diameters having perpendicular ordinates, which are called the principal 

 axes of the ellipsoid. 



Similarly, there are always three directions in the compressed particle in which there 

 is no tangential 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. 



NOTE 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. 



2nd. Regnault's experiments on hollow spheres shew that both linear and cubic com- 

 pressions are proportional to the pressures. 



3rd. Experiments on the elongation of rods and tubes immersed in water, prove that 

 the elongation, the decrease of diameter, and the increase of volume, are proportional to the 

 tension. 



4th. In Coulomb's balance of torsion, the angles of torsion are proportional to the 

 twisting forces. 



It would appear from these experiments, that compressions are always proportional to 

 pressures. 



Professor Stokes has expressed this by making one of his coefficients depend on the 

 cubical elasticity, while the other is deduced from the displacement of shifting produced by 

 a given tangential force. 



M. Cauchy makes one coefficient depend on the linear compression produced by a force 

 acting in one direction, and the other on the change of volume produced by the same force. 



Both of these methods lead to a correct result ; but the coefficients of Stokes seem to 

 have more of a real signification than those of Cauchy; I have therefore adopted those of 

 Stokes, using the symbols m and p, and the fundamental equations (4) and (5), which define 

 them. 



