Page:BatemanConformal.djvu/8

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1908.]
77
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]

If

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

Putting

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