STRAINED ELASTIC SOLIDS 471 



before. In consequence [375] and [376] are true in conditions 

 other than those of equiUbrium; they express in fact, as Gibbs 

 says, "necessary relations," — necessary, that is, in the sense that 

 otherwise there would be involved a contradiction with the 

 laws of dynamics in situations more general than those con- 

 sidered in the text. 



The equations [381] should be compared with (29a) of this 

 article, in which the expression {aXx + fiXy + yXz)Ds is 

 the stress-action across Ds in the direction OX of surface 

 matter on interior matter, and — apDs is F^Ds, the a;-compo- 

 nent of the external force on Ds. The difference here is purely 

 formal, since (a'Xx' + ^'Xy' + y'Xz')Ds' is still the stress- 

 action of surface matter on internal matter across the same 

 element of area which was Ds' in the state of reference. The 

 transformation of the equations to the form [382], which in- 

 volves throughout the direction cosines a', ^', y' of the element 

 in its state of reference, can be obtained at once without going 

 through the argument in Gibbs, I, 192, 193; for we have 

 already considered that argument in somewhat greater de- 

 tail when proving equations (18) and (27). The notation we 

 used in our discussion allows us to write equations [382] more 

 fully, thus, 



a'Xx' + /3'Xr + y'Xz, + p{a'An + /8'^i2 + y'A,^} = 0, 



and two similar equations, since by (27) 

 Ds ( Ka\ 



and An is the second minor of On in the determinant | a\, i.e., 



All = 0,22(133 — 023^32; 



dy dz dz dy 

 ^ dy' dz' ~ dy' dz'' 



and so on. 



We pass on to the arguments based on equation [386] or 

 [387]. The symbols p and mi refer of course to the surrounding 



