to some simple law. We may find another, and perhaps a simpler illustration as follows:
In a very simple and beautiful theorem, Galileo shewed that, if we imagine a number of inclined planes, or gutters, sloping downwards (in a vertical plane) at various angles from a common starting-point, and if we imagine a number of balls rolling each down its own gutter under the influence of gravity (and without hindrance from friction), then, at any given instant, the locus of {563} all these moving bodies is a circle passing through the point of origin. For the acceleration along any one of the sloping paths, for instance AB (Fig. 287), is such that
Therefore
That is to say, all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from A to C.
Where, then, as often happens, the generating curve of the shell is apôÙproxôÙiôÙmateôÙly a circle