Page 266 - Determinants and Their Applications in Mathematical Physics

6.4 The Kay–Moses Equation 251

= b i c i φ i [(c i + c j )A − φ i φ j ]
ij
i

= b r c r φ r [(c r + c j )A rj − φ r φ j ],
r

b r c r A φ = b r c r φ r [c j A rj − φ r φ j ], (6.4.12)
rj
r
r r
2

b r c r A (φ − φ r )= b r c r φ r (c j − 1)A rj − b r c r φ φ j .
rj
r r
r r r
Multiply by e c j u /(c j − 1), sum over j, and refer to (6.4.7):

rj c j u
b r c r A e (φ − φ r ) 2 2 e c j u
= b r c r φ − b r c r φ
r φ j
c j − 1 r r c j − 1
j,r r r j
2
= F b r c r φ
r
r
1

= F(log A) , (6.4.13)
2
where
e c j u
F =1 − φ j
c j − 1
j
e (c i +c j )u A ij
=1 − . (6.4.14)
c j − 1
i,j
Differentiate and refer to (6.4.9):
e c j u A
rj

F = −2 e c i u A ir
c j − 1
b r c r
r
j i
φ r e c j u A rj

= −2 . (6.4.15)
c j − 1
b r c r
r j
Differentiate again and refer to (6.4.8):
e c j u 2
F =2 φ φ j − c j φ r A rj − φ A rj

c j − 1 r r
b r c r
r
j
= P − Q − R, (6.4.16)
where
e c j u 2
P =2 φ j b r c r φ
c j − 1 r
j r
=(1 − F)(log A) (6.4.17)
b r c r c j φ r e c j u A rj
Q =2
c j − 1
j,r