1888.] 



On JEolotropic Elastic Solids. 



215 



it goes. It is employed in solving the problem, already treated by 

 Saint- Venant, of a beam, whose length is perpendicular to the plane 

 of symmetry, held at one end, and at the other acted on by a system 

 of forces, whose resultant consists of a single force along the axis of 

 the beam, and of a couple about any line in the terminal section 

 through its centroid. The cross-section may be any whatever, 

 including the case of a hollow beam, provided it be uniform through- 

 out. The case when the cross- section is elliptical, and the beam 

 exposed to equilibrating torsional couples over its ends is also treated. 

 Results are obtained confirmatory of Saint- Venant's. They are also 

 extended to the case of a composite cylinder, formed of shells of 

 different materials whose cross-sections are bounded by concentric 

 similar and similarly situated ellipses, the law of variation being the 

 same for all the elastic constants of the solution. The limiting case 

 of a continuously varying structure is deduced. 



It is found when a beam is exposed to terminal traction, whether 

 uniform or not, that the strain consists in part of a shear in the plane 

 of the cross-section which is proportional to the traction ; and the 

 position of the lines in the cross-section, which being originally at 

 right angles remain so, is determined. These lines are called principal 

 axes of traction. If there are in addition two planes of symmetry 

 through the axis of the beam, these principal axes are the intersec- 

 tions of the planes of symmetry with the cross-section. 



When a beam of circular section is exposed to torsion, it is proved 

 that warping will ensue proportional to the moment of the twisting 

 couple. Only two diameters in the cross-section, and these mutually 

 at right angles, remain perpendicular to the axis of the beam. These 

 are called principal axes of torsion. If w denote displacement parallel 

 to the axis of the beam, and r, (p denote the undisturbed polar co-ordi- 

 nates of a point in the cross-section, referred to its centre as 

 origin, and one of these axes as initial line, the law of warping is 

 given by — 



w oc r 2 sin 2 0. 



There is in general no connexion between the positions of the prin- 

 cipal axes of traction and of torsion, as the expressions giving their 

 inclination to the axes of co-ordinates contain wholly different elastic 

 constants ; but for three-plane symmetry of the kind already men- 

 tioned they coincide. When the material is symmetrical round the 

 axis of the beam, the shear and the warping of course are found to 

 vanish. It is pointed out how by means of these various properties 

 the nature of the material may be investigated experimentally. 



Part II treats of a material symmetrical round an axis, that of z, 

 and having the perpendicular plane one of symmetry. A general 

 solution of the internal equations of equilibrium is obtained, sup- 



