414 RICE 



ART. K 



X', Y', Z' where X' = K'a', Y' = K'^', Z' = K'y'. Similarly a 

 point whose coordinates are X", Y", Z", where X" = K"a" , 

 etc., can represent the triangle P"Q]"Qi". The equations (18) 

 can then be written 



X" = EnX' + EnY' + ^i3Z',1 



Y" = EnX' + EnY' + EnZ',)- (19) 



Z' = EziX' -\- E32Y' + EzzZ' .^ 



The reader will probably feel intuitively that, as can be estab- 

 lished by definite proof, by choosing the principal axes of strain 

 as the axes of reference, we can reduce the nine coefficients to a 

 form in which £'23 + £'32, -£^31 + E^, En + £'21 are zero, and 

 En, E21, E33 become the principal ratios of superficial enlarge- 

 ment, i.e., TiTs, rsn, viVi. Squaring and adding the equalities 

 in (19) we obtain 



K"^ = EiX" + E2Y'^-\-EzZ'^-\-2EiY'Z' + 2E,Z'X'-\-2EeX'Y', 



where 



£1 = En' + £21^^ + £31^ 



and two similar equations, 

 Ei = Eiibjiz ~r E^itjiz "T Ezitiizz 

 and two similar equations.^ 



(20) 



An application of the theorem in the Mathematical Note already 

 used would lead to the result that the value of Ei-\- E2-\- E3 is 

 independent of the choice of axes (just as was ei + 62 + es in 

 the discussion of equations (3) and its results). Since, with 

 the choice of the principal axes of strain, the values of the Er, 

 are as stated above, it follows that 



£1 + £2 + £3 = (r^ny + (nny + (nr^y, 



which is just equation (15). The details of the proof of these 

 statements are not difficult to supply, but for our purpose it is 

 the result (18) which is important. 



As a final step in the elucidation of Gibbs, I, pages 205-211 

 we shall now adopt Gibbs' plan of allowing the axes to 

 which we refer the system in its strained state to be any set of 



