hooke's law. 39 



mental equations are always derived in this way. and the stress is taken 

 primarily as the force per unit area of the mass in a state of equilibrium. 

 Tims, a less ambiguous statement of this law would be: Stress in an 

 elastic mass which has reached a condition of equilibrium is proportional 

 to the strain which the mass has undergone. 



It is a curious fact that this is not the law which Hooke intended to 

 express. Hooke's words arc " JJt tensio sic vis: That is, the Power of any 

 Spring is in the same proportion with the tension thereof: That is, if one 

 power stretch or bend it one space, two will bend it two, and three will 

 bend it three, and so forward."* Thus Hooke's law as he meant it is 

 clearly load is proportional to strain, and he had no idea of confining his 

 law to infinitesimal deformations. 



When the stresses and strains are infinitesimal it is easy to show that 

 the two assertions, stress is proportional to strain and loadis proportional 

 to .-train, are really equivalent : but for finite deformations they lead to 

 very different results. 



Let a unit cube he extended to a length 1 + e by a load L, and let the 

 reduced area of the cross-section he J. Then the tension per unit area 

 or the stress P is given by — 



L = AP, 



and if stress is proportional to strain, 



p= Me, or L = AMe, 



where M is the constant, called Young's modulus and sometimes (though* 

 improperly) the modulus of elasticity. As was shown above, exactly 

 one-third of the load is-employed in producing dilation, however great 

 L may he. Hence if/.' is the modulus of compressibility, the volume of 

 the distorted cube is 1 /. '■'> !:. The volume is also the area of the dis- 

 torted mass multiplied by its length, or J. (1 + e). Thus — 



, 1 + LIB /■ 



Substituting this value in the last equation gives an equation between 

 Load and strain, viz : 



L — Me + Le 3 k ~ M = 0, 

 ■ > fc 



♦ Quoted by P. G. Tait, " Properties of Matter," 1890, p. 204, from Hooke's lectures " de Potentia 

 Restitutiva." 



VII— Bum,. Gkol. Soc. Am., Vot. 4, 1892. 



