passing through the point of origin, we may consider the acceleration of growth along various radiants to be governed by a simple mathôÙeôÙmatôÙiôÙcal law, closely akin to that simple law of acceleration which governs the movements of a falling body. And, mutatis mutandis, a similar definite law underlies the cases where the generating curve is continually elliptical, or where it assumes some more complex, but still regular and constant form.
It is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical. In such a case we have xÿ£¢2ã₤ãã₤aÿ£¢2ã₤+ã₤yÿ£¢2ã₤ãã₤bÿ£¢2 =ã₤1, where a and b are the major and minor axes of the ellipse.
Or, changing the origin to the vertex of the figure
giving
Then, transferring to polar coordinates, where rã₤ôñã₤cosã₤ö¡ =ã₤x, rã₤ôñã₤sinã₤ö¡ =ã₤y, we have
which is