THE EQUILIBRIUM OF ELASTIC SOLIDS. 



considering the relation of the coefficients which occur in these two cases; 

 while others have treated of the general problem of a solid body exposed to 

 any forces. 



The investigations of Leibnitz, Bernoulli, Euler, Varignon, Young, La Hire, 

 and Lagrange, are confined to the equilibrium of bent rods; but those of 

 Navier, Poisson, Lanit? and Clapeyron, Cauchy, Stokes, and Wertheim, are 

 principally directed to the formation and application of the general equations. 



The investigations of Navier are contained in the seventh volume of the 

 Memoirs of the Institute, page 373 ; and in the Annales de Chimie et de 

 PI' Unique, 2' S^rie, xv. 264, and xxxvm. 435 ; U Application de la Mecanique, 

 Tom. i. 



Those of Poisson in M6m. de I'Institut, vni. 429 ; Annales de Chimie, 2." 

 Serie, xxxvr. 334; xxxvu. 337; xxxvrn, 338; XLII. Journal de I'Ecole 

 Polytechnique, cahier xx., with an abstract in Annales de Chimie for 1829. 



The memoir of MM. Lamd and Clapeyron is contained in Crelle's Math<- 

 matical Journal. Vol. vn. ; and some observations on elasticity are to be found 

 in Lamp's Cours de Physique. 



M. Cauchy's investigations are contained in his Exercices d? Analyse, Vol. m. 

 p. 180, published in 1828. 



Instead of supposing each pressure proportional to the linear compression 

 which it produces, he supposes it to consist of two parts, one of which is pro- 

 portional to the linear compression in the direction of the pressure, while the 

 other is proportional to the diminution of volume. As this hypothesis admits 

 two coefficients, it differs from that of this paper only in the values of the 

 coefficients selected. They are denoted by K and k, and K=p. ^m, k = m. 



The theory of Professor Stokes is contained in Vol. vni. Part 3, of the 

 Cambridge Philosophical Transactions, and was read April 14, 1845. 



He states his general principles thus : " The capability which solids possess 

 of being put into a state of isochronous vibration, shews that the pressures 

 called into action by small displacements depend on homogeneous functions of 

 those displacements of one dimension. I shall suppose, moreover, according to 

 the general principle of the superposition of small quantities, that the pressures 

 due to different displacements are superimposed, and, consequently, that the 

 pressures are linear functions of the displacements." 



