J. W. Gibhs — Equilihrinin of Heterogeneous ^ubxtatices. :{(35 



^=^(S)'' ^=^(:^r- -=^(ir. <-) 



^./dx dx\ - ^,/dxdx\ ^ fdx dx\ 



«=^W&'> *=^W&'> ^•=^(&'*7> <*'") 



The determination of the fundamental equation for a solid is thus 

 reduced to the determination of the relation between fy,, ;/v,, A^ 7>, 

 G, a, 5, c, or of the relation between ^\, , t, A, _S, C, «, b, <: 



In the case of isotrojiic solids, the state of strain of an element, so 

 far as it can affect the relation of s^;, and i}y, , or of //'y, and t, is capa- 

 ble of only three independent variations. This appears most dis- 

 tinctly as a consequence of the proposition that for any given strain 

 of an element there are three lines in the element which are at right 

 angles to one another both in its unstrained and in its strained 

 state. If the unstrained element is isotropic, the ratios of elonga- 

 tion for these three lines must with //y, determine the value fy,, or 

 with t determine the value of //'y, . 



To demonstrate the existence of such lines, which are called the 

 principal axes of strain, and to find the relations of the elongations 



of such lines to the quantities — , , . . . -^ , we may proceed as fol- 

 lows. The ratio of elongation ;• of any line of which a', /^', y' are 

 the direction-cosines in the state of reference is evidently given by 

 the equation 



, /dz , , dz .-., clz \" , ^ 



Now the proposition to be established is evidently equivalent to this 

 — that it is always possible to give such directions to the two sys- 

 tems of rectangular axes JC', Y', Z', and JT, y, Z, that 



dx dx dy ^ 



^'^^' ^=^' dz'^"^' I 



dy dz dz \ 



£'="' &■■="' *7'="-j 



We may choose a line in the element for which the value of r is at 

 least as great as for any other, and make the axes of JC and X' par- 

 allel to this line in the strained and unstrained states respectively. 



