416 



RICE 



ART, K 



Just as before, we recognize that the determinant in (24) is the 

 square of the determinant 



(25) 



and this is actually the determinant indicated by H in Gibbs, 

 while the one in (24) is there indicated by G. Hence equations 

 [437] and [442] are included in (24) and (25). We have been 

 using a double suffix and single suffix notation as the most 

 convenient to follow in this exposition and the most consis- 

 tent with present day practice, but for comparison with Gibbs' 

 treatment the reader will observe that A, B, C, a, h, c defined by 

 him in [418] and [419] are respectively ai, a2, az, ai, as, a^ in this 

 exposition. A glance shows that the first of equations (24) is the 

 first of the equations [439]. The second of (24) is, as before, 

 a little more troublesome to deal with by straightforward algebra, 

 but it can be verified that the expression on the left hand side 

 is the sum of the squares of the nine first minors of the determi- 

 nant (25) . A similar notation for these minors can be introduced 

 as before, viz., An for the minor of an, A^ for that of ai2, A^i 

 for that of 021, etc. Thus the equations (24) can be written 



/ J 2j ^ra^ = ^1^ + ^2^ + ^3^ 

 r s 



au ^12 ^13 



O21 Ct22 023 

 O31 O32 033 



2 



= r^r^r-^. 



(26) 



The left hand side of the first of these is the expression denoted 

 by E in Gibbs; the expression on the left hand side of the 

 second is referred to as F (see [432] and [434]), and, as already 

 mentioned, H is used for the determinant in (25) and G for 

 the determinant in (24). Thus equations (26) are just the 



