444 RICE 



ART. K 



exist between Yx' and Xy, and so on, provided the two sets of 

 axes are regarded as coincident. 



At the risk of appearing to be prohx on this matter, the writer 

 would hke to point out that the equations (38) offer an alter- 

 native method of giving the correct physical signification to 

 Xx', etc. If we recall the arguments developed from equations 

 (16) to (27) above, we will remember on looking at (27) that a 

 unit area, which was in the state of reference perpendicular to 

 OX' (so that for it a' = 1, jS' = 0, 7' = 0), is strained into an 

 area whose projections on the planes perpendicular to OX, OY, 

 OZ are An, A21, Asi, with similar results for unit areas originally 

 normal to OY' or OZ'. In other words, if unit area which was 

 in the state of reference perpendicular to OX' is strained into an 

 area of size K with direction cosines a, j8, 7 with reference to 

 OX, OY, OZ, then 



Ka = An, 

 m = A21, 

 Ky = A31. 



But by (28), the force across this surface in the state of strain in 

 the direction OX has the value aXx + ^Xy + 7X2 per unit 

 area, and so the actual force across the area Kin the state of 

 strain is 



AnXx + A21XY + AsiXz, 



which by (38) is just Xx', thus giving us the physical inter- 

 pretation of Xx' once more. In the same way we can demon- 

 strate that Xy' is the force parallel to OX across an area in the 

 state of strain, which in the state of reference was unit area in 

 size and normal to OY' in orientation; and so on. 



6. Thermodynamics of a Strained Homogeneous Solid. The 

 treatment of heterogenous systems in the earlier parts of Gibbs' 

 discussion of the subject is of course based on equation [12] which 

 is a generalization from equation [11], the equation for a 

 homogeneous body when uninfluenced by distortion of solid 

 masses (among other physical changes). In the same way any 

 treatment of heterogenous substances in which elastic effects 



