AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 315 



To show how implicitly the common theory of elasticity seems to be received by some, I may 

 mention that MM. Lame and Clapeyron mention Indian rubber among the substances to which 

 it would seem they consider their theory applicable *. I do not know whether the coefficient of 

 elasticity, according to that theory, has been determined experimentally for Indian rubber, but 

 one would fancy that the cubical compressibility thence deduced, by a method which will be 

 seen in the next article, would turn out comparable with that of a gas. 



20. I am not going to enter into the solution of equations (30), but I wish to make a 

 few remarks on the results in some simple cases. 



If k be the cubical contraction due to a uniform pressure P, then will 



-?■ 



If a wire or rod, of which the boundary is any cylindrical surface, be pulled in the 

 direction of its length by a force of which the value, referred to a unit of surface of a 

 section of the rod, in P, the rod will extend itself uniformly in the direction of its length, 

 and contract uniformly in the perpendicular direction; and if e be the extension in the 

 direction of the length, and c the contraction in any perpendicular direction, both referred to 

 a unit of length, we shall have 



A + B _ A-2B 



^~ SAP ^' ''~'6AB~ 



P 



also, the cubical dilatation = e — 2 c = — . 



A 



If a cylindrical wire of radius r be twisted by a couple of which the moment is M, and 

 if be the angle of torsion for a length z of the wire, we shall have 



e = ^. 



■KBr" 



The expressions for k, c, e and d, and of course all expressions of the same nature, depend 

 on the reciprocals of A and B. Suppose now the value of. e, or Q, or any similar quantity 

 not depending on A alone, be given as the result of observation. It will easily be conceived 

 that we might find very nearly the same value for B whether we supposed A = 5B or A = nB, 

 where n may be considerably greater than 5, or even infinite. Consequently the observation of two 

 such quantities, giving very nearly the same value of B, might be regarded as confirming the 

 common equations. 



If we denote by E the coefficient of elasticity when A is supposed to be equal to 5 B 

 we have, neglecting the atmospheric pressure-|-, 



2P ^ iMz 



e = , 6 = ; . 



5E TrEr* 



If now we denote by £, the value of E deduced from observation of the value of e, and by 

 A'j the value of E obtained by observing the value of 9, or else, which comes to the same, 

 by observing the time of oscillation of a known body oscillating by torsion, we shall have 



5-i. =^(i + Zf)' ^-=^' -hence. = 



I __6 1^ 



A ~ JE, Ei ' 



' Mlmoirei pritenUt a rjntlilul, Tom. iv. p. 469. f Lame, fours de I'lij/siqiie, Tom. 



S S 2 



