16 Sir W. Thomson on the Thermoelastic, Thermomagnetic, 



ner the conditions of equilibrium of a solid in any state of strain 

 whatever at a constant temperature. They show how the 

 straining forces are altered with any infinitely small alteration 

 of the strain. If we denote by P &c. the values of P &c. for 

 the state (a , yo? z o, ?o, Vo, 5» 0? ^ e values of P — P , Q — Q , 

 R— R , S — S , T— T , U — U given by these equations as 

 linear functions of the strains (x— x ), (y— y ), (z — z ), (f — f ), 

 (v— lo)? (£— &>), with twenty-one coefficients, express the whole 

 tensions required to apply these strains to the cube, if the con- 

 dition of the solid when the parallelepiped is exactly cubical 

 is a condition of no strain, and in this case become (if single 



letters are substituted for the coefficients I -j-^ ) &c.) identical 



with the equations of equilibrium of an elastic solid subjected 

 to infinitely small strains, which have been given by Green, 

 Cauchy, Haughton, and other writers. Many mathematicians 

 and experimenters have endeavoured to show that in actual 

 solids there are certain essential relations between these twenty- 

 one coefficients [or moduluses] of elasticity. Whether or not it 

 may be true that such relations do hold for natural crystals, it is 

 quite certain that an arrangement of actual pieces of matter may 

 be made, constituting a homogeneous whole when considered on 

 a large scale (being, in fact, as homogeneous as writers adopt- 

 ing the atomic theory in any form consider a natural crystal to 

 be;, which shall have an arbitrarily prescribed value for each 

 one of these twenty-one coefficients. No one can legitimately 

 deny for all natural crystals, known and unknown, any pro- 

 perty of elasticity, or any other mechanical or physical pro- 

 perty, which a solid composed of natural bodies artificially put 

 together may have in reality. To do so is to assume that the 

 infinitely inconceivable structure of the particles of a crystal is 

 essentially restricted by arbitrary conditions imposed by mathe- 

 maticians for the sake of shortening the equations by which their 

 properties are expressed. It is true experiment might, and does, 

 show particular values for the coefficients for particular bodies ; 

 but I believe even the collation of recorded experimental inves- 

 tigations is enough to show bodies violating every relation that 

 has been imposed ; and I have not a doubt that an experi- 

 ment on a natural crystal, magnetized if necessary, might be 

 made to show each supposed relation violated. Thus it has 

 been shown, first I believe by Mr. Stokes, that the relation 

 which the earlier writers supposed to exist between rigidity 

 and resistance to compression is not verified, because experi- 

 ments on the torsion of wires of various metals, rods of india- 

 rubber, &c. indicate, on the whole, less rigidity than would be 

 expected, according to that relation, from their resistance to 



