Page:BatemanConformal.djvu/8

1908.] 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.

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