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The conformal transformations of a space of four dimensions.

a solution of

the function

is also a solution.

In following up the connection between different solutions, it is convenient to use polar coordinates. Putting

we obtain the relations

There is a similar transformation for Laplace's equation.[1]


a solution f(x, y, z) corresponds to a second solution


the formulæ of transformation become

The transition from one solution of Laplace's equation to another is now easily effected.

The effects of combining the different transformations belonging to a group of conformal transformations is most easily studied by interpreting

  1. This transformation was given by the author in a Smith's Prize Essay of 1905; it was deduced from a result given by Brill, Messenger of Mathematics (1891), pp. 135-137. If is a solution of the differential equation

    another solution is given by