497 



Biquadratic Tasinomic 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 thxis 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 order 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 employment along with them of three auxiliary 

 variables called Contra-ordinates. The contra-ordinates of a point 

 R are the projections of the radius-vector OR on the three axes. 

 For rectangular axes, co-ordinates and contra-ordinates are identical. 

 The co-ordinates x, y, z and contra-ordinates , t;, w of a point R 

 are connected by the equation 



ux + vy + wz = OR 2 . 



As there are six independent quantities in the directions of a 

 system of three axes of indefinite obliquity, there is a system of 



