14: Sir W. Thomson on the lliermoelastic, Thermomagnetic, 



14. According to the system of variables* which we have 

 adopted, as set forth in § 12, when x—x &c. are each infinitely- 

 small, x increasing corresponds to a motion of all the particles 

 in a plane at a distance unity from YOZ, in directions perpen- 

 dicular to this plane, through a space numerically equal to the 

 increment of x ; f increasing corresponds to a motion of all the 

 particles at a distance unity from XOY, in directions parallel to 

 YO, through a space equal to the increment of £, or to a motion 

 of all the particles at a distance unity from XOZ, in directions 

 parallel to ZO, through a space equal to the increment of f , or 

 to two such motions superimposed, through any spaces respec- 

 tively, amounting together to a quantity equal to the incre- 

 ment of f . Similar statements apply to the effects of variations 

 of the other four variables. Hence, if P, Q, R denote the nor- 

 mal components of the superficial tensions experienced respec- 

 tively by the three pairs of opposite faces of the unit cube of 

 the solid in the state of strain in which we are considering it, 

 and if S, T, U be the components, along the planes of the faces, 

 of the actual tensions, taken in order of symmetry, so that S 

 denotes the component, perpendicular to the edge opposite to 

 OX, of the superficial tension in either of the faces meeting in 

 that edge (which are equal for these two faces, or else the cube 

 would not be in equilibrium, but would experience the effect 

 of a couple in a plane perpendicular to OX), and T and U de- 

 note components, perpendicular respectively to OY and OZ, 

 of the superficial tensions of the pairs of faces meeting in those 

 edges, the work done on the parallelepiped during an infi- 

 nitely small strain in which the variables become augmented 

 by dx, dy y &c. respectively will be 



Ydx + Qdy + Rdz + Sdf + Tdy + Urf£ 

 Hence, if the portion of matter of which the intrinsic energy 

 is denoted by e, and to which the notation e, w, <fcc. applies, be 

 the matter within the parallelepiped referred to, we have 



dw __-p dw 



dw a dio ^ dw TT j 

 df ' A? ' W J 



15. Using the development of w expressed by (18), we derive 

 from these equations the following expressions for the six 

 component tensions : — 



* [A method of generalized stress and strain components is fully de- 

 veloped in "Elements of a Mathematical Theory of Elasticity/' first pub- 

 lished in the Transactions of the Royal Society for April 1856, and 

 embodied in an article on "Elasticity," about to be published in the 

 Encyclopcedia Britannica.~] 



aw -p, aw ~ dw -o 1 



-v-=P, -r-=Q, ^=&, I 



