12 Sir W. Thomson on the Therm oelastic, Thermomagnetic, 



■m 



m ~ m ~-jtf ■ ■ ■ ■ <"> 



dv 

 and d(JK) _ <Pp_ ( . 



~~d^~ - <#' {) 



which are the equations used to express those relations in a 

 recent paper by Mr. Joule and myself, " On the Thermal 

 Effects of Fluids in Motion" *. 



If, instead of , we substitute fidt, considering fi as a 



V 



function of Carnot's function of the temperature, they be- 

 come identical with the two fundamental equations (14) and 

 (16) given in Part III. of my first communication " On the 

 Dynamical Theory of Heat" f. 



11. To apply the preceding equations to a body possessing 

 rigidity, it is necessary to take the form as well as the bulk into 

 account, and therefore to retain, besides the temperature, six 

 independent variables to express those elements. There is, of 

 course, an infinite variety of ways in which the form and bulk 

 of a homogeneously strained body may be expressed by means 

 of six independent variables. Thus the lengths (three vari- 

 ables) and the mutual inclinations (three variables) of the 

 edges of a parallelepiped enclosing always the same portion of 

 the solid in all states of strain (which of course always remains 

 a parallelepiped, provided the strain is homogeneous through- 

 out the solid), may be chosen for the independent -variables ; 

 or we may choose the six elements of an ellipsoid enclosing 

 always the same portion of the solid (which will always remain 

 an ellipsoid however the solid be strained, provided it is strained 

 homogeneously. Thus, let us actually take for x, y, z the lengths 

 of three conterminous edges OX, OY, OZ of a certain paral- 

 lelepiped of the solid, and for f , rj, f the angles between the 

 planes meeting in these edges respectively, the parallelepiped 

 being so chosen that it becomes strained into a cube of unit 

 dimensions, when the solid is in the particular state at which 

 we wish to investigate its thermoelastic properties. 



12. If then we take 



#0 = 1? yo = I; z 0~*-) 



and if we suppose x, y, z, f , y, % to differ infinitely little from 

 x , y , z , f , rj 9 , f respectively, the actual state (x, y, z, f , rj, £) 

 will be one in which the body is strained from the state 



* Transactions of the Eoyal Society, June 15, 1854. 



t Transactions of the Royal Society of Edinburgh, March 17, 1851. 



