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6.2 Brief Historical Notes 243

which yields only two independent scalar equations, namely

1 2 2 2 2
− φ − φ + ψ + ψ =0, (6.2.8)
ρ ρ z ρ z
φ φ ρρ + φ ρ + φ zz

1
− 2(φ ρ ψ ρ + φ z ψ z )=0. (6.2.9)
ρ
φ ψ ρρ + ψ ρ + ψ zz
The second equation can be rearranged into the form
∂
∂
+ ρρ z =0.
ρρ ρ
∂ρ φ 2 ∂z φ 2
Historically, the scalar equations (6.2.8) and (6.2.9) were formulated before
the matrix equation (6.2.1), but the modern approach to relativity is to
formulate the matrix equation first and to derive the scalar equations from
them.
Equations (6.2.8) and (6.2.9) can be contracted into the form
2
2
2
φ∇ φ − (∇φ) +(∇ψ) =0, (6.2.10)
2
φ∇ ψ − 2∇φ · ψ =0, (6.2.11)
which can be contracted further into the equations
1 (ζ + + ζ − )∇ ζ ± =(∇ζ ± ) , (6.2.12)
2
2
2
where
ζ + = φ + ωψ,
2
ζ − = φ − ωψ (ω = −1). (6.2.13)
The notation
ζ = φ + ωψ,
ζ = φ − ωψ, (6.2.14)
∗
where ζ is the complex conjugate of ζ, can be used only when φ and ψ are
∗
real. In that case, the two equations (6.2.12) reduce to the single equation
1 (ζ + ζ )∇ ζ =(∇ζ) . (6.2.15)
2
2
∗
2
In 1983, Y. Nakamura conjectured the existence two related infinite sets of
solutions of (6.2.8) and (6.2.9). He denoted them by

φ ,ψ , n ≥ 1,

n n
φ n ,ψ n , n ≥ 2, (6.2.16)
and deduced the first few members of each set with the aid of the pair of
coupled difference–differential equations given in Appendix A.11 and the
B¨acklund transformations β and γ given in Appendix A.12. The general
Nakamura solutions were given by Vein in 1985 in terms of cofactors as-
sociated with a determinant of arbitrary order whose elements satisfy the