302 Royal Society :— 



The existence of ortliotatic axes in a solid constituted of mutu- 

 ally attracting and repelling physical points was first proved by 

 Mr. Haughton ; it is proved in this paper independently of any 

 hypothesis as to molecular structure or action. 



Heterotatic Axes. 



At each point of an elastic solid there is one position in which a 

 cubical molecule may be cut out, such, that if there be a distortion of 

 that molecule round x (x being any one of its axes) and an equal dis- 

 tortion round y (y being either of its other two axes), the normal stress 

 on the faces normal to x arising from the distortion round x, will be 

 equal to the tangential stress around z arising from the distortion 

 rotind y. 



The six coefficients of the Heterotatic Ellipsoid represent parts of 

 the elasticity of a solid which it is impossible to reduce to attrac- 

 tions and repulsions between points. 



Fifteen constants called the Homotutic Coefficients, which are com- 

 posed of Tasinomic Coefficients and their linear functions so con- 

 stituted as to be independent of the Heterotatic Coefficients, are the 

 coefficients of the fifteen terms of a homogeneous biquadratic function 

 of the co-ordinates, which being equated to unity, characterizes the 

 Biquadratic Tasinom.ic Surface. This surface, for solids composed 

 of physical points, was discovered by Mr. Haughton; it is here in- 

 vestigated independently of all hypothesis. 



By rectangular linear transformations, three functions of the 

 Homotatic Coefficients may be made to vanish. Three orthogonal 

 axes are thus found, which are called the Principal Metatatic Axes, 

 and have the following property : if there be a linear elongation 

 along any one of these axes, and an equal linear compression along 

 any other, no tangential stress will result on planes normal to these 

 two axes. 



In each of the three planes of the principal Metatatic Axes, there 

 is a pair of Diagonal Metatatic Axes bisecting the right angles 

 formed by the pair of principal axes in the same plane. 



In each plane in an elastic solid, there is a system of two pairs of 

 metatatic axes, making with each other eight equal angles of 45°. 



Various kinds and degrees of symmetry are pointed out, which the 

 tasinomic coefficients may have with respect to orthogonal axes. 



The Potential Energy of Elasticity may be expressed as a homo- 

 geneous function of the second ordt-r of the Elementary Stresses. 

 The twenty- one coefficients of this function are called Thlipsinomic 

 Coefficients. 



The Thlipsinomic and Tasinomic Coefficients are related to each 

 other as Contragredient Systems. 



The Orthogonal and Diagonal Tasinomic and Thlipsinomic Axes 

 coincide. 



For the complete determination of the properties of the Homo- 

 tatic Coefficients, it is necessary to refer them to oblique axes of co- 

 ordinates. 



The application of oblique co-ordinates to this purpose is much 

 facilitated by the emijloyment along with them of three auxiliary 



