Page:BatemanConformal.djvu/10

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1908.]
79
The conformal transformations of a space of four dimensions.

where P is a function of the ratio of any two of the quantities lλ, mμ, nν. The quantity U will then be a solution of the equation

.

if the relation

is satisfied, for it will then be a homogeneous function of degree -1 in (l, m, n, λ, μ, ν).

Let us put

where a, b and c are arbitrary constants. The relation

is then satisfied, and P becomes a function of z alone. We may thus write

the particular functional form in terms of ξ, η, ζ being chosen to facilitate the calculations. H is clearly a homogeneous function of degree zero in ξ, η, ζ and therefore in l, m, n, λ, μ, ν.

On differentiating equation (1), we obtain

The differential equation

will thus be satisfied, if