Page:Newton's Principia (1846).djvu/237

$\scriptstyle {\frac {2SI^{3}}{3LI}}$. And these by subducting the last from the first, become $\scriptstyle {\frac {4SI^{3}}{3LI}}$. Therefore the entire force with which the corpuscle P is attracted towards the centre of the sphere is as $\scriptstyle {\frac {SI^{3}}{PI}}$, that is, reciprocally as PS³ $\scriptstyle \times$ PI.   Q.E.I.

By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following Theorem.

PROPOSITION LXXXII. THEOREM XLI.

In a sphere described about the centre S with the interval SA, if there be taken SI, SA, SP continually proportional; I say, that the attraction of a corpuscle within the sphere in any place I is to its attraction without the sphere in the place P in a ratio compounded of the subduplicate ratio of IS, PS, the distances from the centre, and the subduplicate ratio of the centripetal forces tending to the centre in those places P and I.

As if the centripetal forces of the particles of the sphere be reciprocally as the distances of the corpuscle attracted by them; the force with which the corpuscle situate in I is attracted by the entire sphere will be to the force with which it is attracted in P in a ratio compounded of the subduplicate ratio of the distance SI to the distance SP, and the subduplicate ratio of the centripetal force in the place I arising from any particle in the centre to the centripetal force in the place P arising from the same particle in the centre; that is, in the subduplicate ratio of the distances SI, SP to each other reciprocally. These two subduplicate ratios compose the ratio of equality, and therefore the attractions in I and P produced by the whole sphere are equal. By the like calculation, if the forces of the particles of the sphere are reciprocally in a duplicate ratio of the distances, it will be found that the attraction in I is to the attraction in P as the distance SP to the semi-diameter SA of the sphere. If those forces are reciprocally in a triplicate ratio of the distances, the attractions in I and P will be to each other as SP² to SA²; if in a quadruplicate ratio, as SP³ to SA³. Therefore since the attraction in P was found in this last case to be reciprocally as PS³ $\scriptstyle \times$ PI, the attraction in I will be reciprocally as SA³ $\scriptstyle \times$ PI, that is, because SA³ is given reciprocally as PI. And the progression is the same in infinitum. The demonstration of this Theorem is as follows:

The things remaining as above constructed, and a corpuscle being in any