ART. K 



486 RICE 



article by another method. (As mentioned at that point a 

 straightforward, if tedious, piece of algebra will show that 



0203 + ascti + aia2 — 04—05 — 



6 



= Al + Al^^... +A 



2 



33' 



where Apg is the first minor of Opg in the determinant of the 

 coefficients, viz. H. This gives the alternative expression for 

 F in [434]. Also, we have already seen that the rule for multi- 

 plying determinants will verify that H^ = G.) A rather special 

 point is raised and disposed of on pages 210, 211. It concerns 

 the sign of the determinant H. It is clear from [439] that G 

 is a positive quantity, but H may, of course, have a negative 

 value instead of a positive one from a purely mathematical stand- 

 point; but from a physical standpoint negative values of H are 

 ruled out, provided we agree that the axes OX', OY', OZ' and 

 OX, OY, OZ are capable of superposition, meaning that if the 

 latter are turned so that OX points along OX', and OY along 

 OY', then OZ will point along OZ' (not along Z'O). In short, if 

 one set of axes is "right-handed" the other must be likewise, 

 if one is ''left-handed," so also is the other. (A right-handed 

 set of axes is one so oriented that to an observer looking in the 

 direction OZ', a right-handed twist would turn OX' to OY', etc.) 

 Gibbs illustrates this by considering a displacement of the 

 particles which is represented by 



X = x', y = y', z = -z', 



the two sets of axes being regarded as identical. (If they were 

 not they could easily be made so by a rotation.) Now the H 

 determinant of this is 



whose value is —1. But such a displacement is one which 

 moves every particle to the position of its "mirror image" with 

 respect to a mirror imagined as located in the plane z' = 0, i.e, 



