and Pyroelectric Properties of Matter. 23 



to the six lines which are lines of symmetry when the solid is 

 unstrained. In any state of strain let x, y, z be the lengths of 

 three edges lying in one plane, and f , 77, £ those of the three 

 others (which meet in a point). Let x Q , y , z , f , rj , ? denote 

 the values (equal among themselves) of these variables for the 

 unstrained state, and let w be the work required to bring any 

 portion of the solid (whether the tetrahedron itself or not is of 

 no consequence) from the unstrained state to the state (x, y, z, 

 f , 7] , J) while kept at a constant temperature. The relations 

 among the coefficients of elasticity according to this system of 

 variables, to express perfect symmetry with reference to the 

 six axes, will clearly be: — 



/d?w\ __.(<Pw\ _ (d?w \ _ (cPw\ _ (<&w\ _ (cPw\ _ 



wj - w) - l¥j fl - \wh~~ W V W A~ OT; 



/ d 2 w \ / d 2 w \ __ f d?iv \ _ ( d?w \ __ / °? w \ 

 \dy dzj ~ \dzdx/ \dxdyJo \drjd^/ \d^d^J Q 



_ f d?w \ __ / d?w \ _ f d?w \ __ f (&w \ 

 \dtj drj) ~~ \dx dr\)~ \dx dy ~ \dy d%) 



_ f d?w \ _ / d 2 w \ __ / d?w \ __ 



= Wdlh \fadSJo~ \QW V ) - a '' 

 / d\v \ __ / dho \ _ f d?iv \ __ 

 V dx d%) ~~\dy dr)) ~ \dz d^) ~~ ' 



where -cr, a, co denote three independent coefficients of elasticity 

 for the substance. The definition of the new system of vari- 

 ables may be given as simply, and in some respects more con- 

 veniently, by referring to the dodecahedron, whose faces are 

 perpendicular to the edges of the tetrahedron. Thus the six 

 variables x, y, z, £, rj, f may be taken to denote respectively the 

 mutual distances of the six pairs of parallel faces of the rhom- 

 bohedron into which the regular dodecahedron is altered when 

 the solid is strained in any manner. Thus, if the portion of the 

 solid considered be the dodecahedron itself, and of such dimen- 

 sions that when it is in its normal state the area of each face is 



unity, the values of -5— &c, denoted, as in the preceding paper, 



by P, Q, R, S, T, U, are normal tensions (reckoned, as usual, 

 per unit of area), on surfaces in the solid parallel to the faces 

 of the dodecahedron, which compounded give the actual strain- 

 ing force to which the solid is subjected. The coefficients 

 denoted above by -cr, cr, co are such as to give the following ex- 

 pressions for the component straining tensions in terms of the 

 strains : — 



