482 RICE 



ART. K 



material. The fr quantities are themselves functions of the six 

 quantities ei, e-i, ... ee (or ai, a2, ... ae) which are the same as 

 A, B, C, a, h, c defined in [418], [419]. Thus the energy or free- 

 energy functions must be functions of these six quantities, or 

 in other words "the determination of the fundamental equation 

 for a solid is thus reduced to the determination of the relation 

 between ev, riv, A, B, C, a, b, c, etc." (page 205). Having 

 pointed this out Gibbs at once proceeds to discuss a further 

 limitation on the form of these functions if the solid is isotropic, 

 and this involves him at once in an appeal to the existence of 

 three principal axes of strain for any kind of material, a fact 

 to which we have already referred in this article. Thereafter 

 he deals with approximations to the form of these functions 

 and concludes this subsection on that topic. 



Let us proceed to the subject matter of pages 205-209 of the 

 original which has been treated in our discussion in a somewhat 

 different manner. The starting point of Gibbs' treatment is the 

 equation [420] and this has already appeared implicitly in this 

 article. For we know that if P' and Q' are the positions in the 

 state of reference of two adjacent points, and P and Q are their 

 positions in the state of strain, then 



PQ' = air' + a2v" + asf" + 2a4Vr' + 2a,^'^' + 2ae^'r,', 



where x', y', z' and x' + ^ , y' + tj', z' + f ' are the coordinates 

 of P' and Q! and ai, ai, az, ai, as, ae are six functions of the 

 strain coefficients defined in (23), or, as already stated, the same 

 functions which Gibbs defines in [418] and [419] denoted by the 

 symbols A, B,C, a, b, c, respectively. If a, ^', y' are the direc- 

 tion-cosines of P'Q' with reference to the axes OX', OY', OZ' 

 so that a' = ^'/P'Q', etc., it follows that 



PQ" 



aia'2 -f- a2)3'2 + asj'^ + 2a,^'y' + 2a,y'a' + 2a6a'^' = =^ = 7- 



P'Q' 



which is just Gibbs' equation [420]. 



The method pursued by Gibbs at this point to demonstrate 

 the existence of the principal axes of strain employs the analyti- 

 cal processes associated with the discovery of maximum- 



