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74 is a solution of

when expressed in terms of x, y, z, w.

Next, let U be a homogeneous function of degree -1 in (l, m, n, λ, μ, ν), then we can show in a similar way that

Hence, if U is a solution of

i.e., of

it is a solution of

when expressed in terms of (x, y, z, w).

When ($$a_{1},\ a_{2},\ \dots,\ a_{6}$$) are interpreted as the coordinates of a point in a space of six dimensions, the expressions

remain unaltered in form after any change of rectangular axes in which the origin remains the same. Any change of this kind corresponds to a transformation in the (x, y, z, w) space, enabling us to pass from one solution of the equation

to another, and a similar remark applies to the equation

To illustrate the method of formation of the transformation, we may consider the effect of simply interchanging n and -ν. The functions V and U of formulæ (8) and (4) then transform into

and

respectively, r² being written in place of x² + y² + z² + w².