The principles of the Calculus of Forms, and in particular the 

 Umbral Notation of Mr. Sylvester, are applied to the Orthogonal 

 Transformations of the Tasinomic Coefficients. 



Several functions of these coefficients are determined, called Tasi- 

 nomic Invariants, which are equal for all systems of orthogonal axes 

 in the same solid. 



Certain functions of the Tasinomic Coefficients constitute the 

 coefficients of two Tasinomic Ellipsoids, designated respectively as 

 the Orthotatic and Heterotatic Ellipsoids, whose axes have the fol- 

 lowing properties. 



ORTHOTATIC AXES. 



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

 cubical molecule may be cut out, such, that a uniform dilatation or 

 condensation of that molecule by equal elongations or compressions of 

 its three axes, will produce no tangential stress at the faces of the 

 molecule. 



The existence of orthotatic 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 

 round 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 Homotatic 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 coordinates, which being equated to unity, characterizes the 



