Page:Scientific Memoirs, Vol. 1 (1837).djvu/468

456 and it being observed that with respect to which see the third volume of the Bulletin de la Société Philomatique, p. 388.

If in this equation we change the differentials taken relatively to the rectangular co-ordinates into differentials taken relatively to the polar co-ordinates, we have

Let us suppose that $$r\,q$$ is developed in a series of integer and rational functions of the spherical co-ordinates, so that we may have in which any one of the quantities $$Q_i$$ renders identical the equation

On this supposition the equation (1) will be satisfied by taking in general

In order to integrate this differential equation of the second order let us take and consequently