Royal Society. 301 



At the period of birth muscular fibres vary much in size. 



The several stages in the development of muscular fibre, above 

 mentioned, do not succeed each other as a simple consecutive series; 

 on the contrary, two, or more, are generally progressing at the same 

 time. Nor does each commence at the same period in all cases. 



" On the General Integrals of the Equations of the Internal 

 Equilibrium of an Elastic Solid." By W. J. Macquorn Rankine, 

 F.R.SS.L. & E. 



The First Section of this paper is introductory, containing a sum- 

 mary of principles already known respecting the elasticity of solids. 

 Those principles are treated as the consequences of the following 

 Definition of Elasticity, without introducing any hypothesis as 

 to the molecular structure of bodies. 



" Elasticity is the property which bodies possess of preserving deter- 

 minate volumes and figures under given pressures and temperatures, and 

 which in a homogeneous body manifests itself equally in every part of 

 appreciable magnitude." 



The investigations are limited by the following conditions : — 



1 . The temperature of the elastic body is supposed to be con- 

 stant and uniform. 



2. The variations of the volumes and figures of its particles are 

 supposed to be so small, that the elastic pressures may be considered 

 as sensibly linear functions of those variations. 



3. It is assumed, that the only force, besides elastic pressures, 

 acting on the particles of the body, is that of terrestrial gravitation. 



All possible small variations of volume and figure of an originally 

 rectangular molecule, when referred to three orthogonal axes, may 

 be resolved into six, viz. three linear dilatations or compressions, 

 and three distortions. 



In like manner the elastic pressures exerted on and by such a 

 molecule may be resolved into six, viz. three normal pressures, and 

 three tangential pressures. 



Those six pressures are connected with each other and with the 

 attractive force acting on the molecule, by three well-known differ- 

 ential equations of the first order. 



They are also connected at every element of the surface of the 

 body, by three well-known linear equations, with the components of 

 the external force acting on that element. 



The general problem to be solved is, to find the integrals of the 

 first three equations, subject to conditions fixed by the last three. 



The six variations of volume and figure of a rectangular molecule 

 are expressed by six small fractions called " coefficients of displace- 

 ment." 



If the differential of each of these fractions be multiplied by the 

 pressure which directly tends to vary it, the sum of the products is 

 the complete differential of a function called the Potential Energy 

 of Elastic Forces for the molecule in question, which is sensibly a 

 homogeneous quadratic function of the six fractions. It has twenty- 

 one terms, and twenty-one constant coefficients, which constitute 

 the Coefficients of Elasticity of the body, for the system of orthogonal 

 axes chosen. 



