ATTRACTION OF Souris. As this subject is so intimately connected with the important experi ments on the attraction of mountains and leaden balls, and with many other branches of physics, and as it cannot be introduced with propriety under any other head, we shall present the reader with some of the most important and useful propositions, referring to other works for the complete discussion of the sub ject.
In the chapter of Physical ASTRONOMY, entitled, On the Gravitation of a Sphere, we have already entered upon the subject as connected with astronomy ; we shall therefore resume the discussion where it was left in that article, following implicitly the steps of New ton, in so far as he has prosecuted the subject in the first book of his Principia. 'We shall then consider the subject of the solids of greatest attraction, which has been recently treated with such ability by Pro fessor Playfair, availing ourselves of the kind permis sion of that distinguished philosopher, to give an abridged view of his valuable paper.
We have already seen, in the article already men tioned, that when the law of the force exerted by the particles is inversely as the square of the distance, the centripetal forces of the spheres themselves, on rece ding from the centre, decrease or increase according to the same law. It will appear front the two follow ing propositions, that when the law of the force va ries in the simple inverse ratio of the distance, the centripetal forces of the spheres in receding from the centre will vary according to the same law as the forces of the particles.
If centripetal forces tend to the several points of spheres, proportional to the distances of those points from the attracted bodies ; the compounded force, with which two spheres will attract each other mu tually, is as the distance between the centres of the spheres.
Case 1. Let AEBF be a sphere ; S its centre • P a particle attracted ; PASB the axis of the sphere passing through the centre of the particle ; EF, cf; two planes, by which the sphere is cut, perpendicular to this axis, and equally distant on each side from the centre of the sphere ; G, g, the intersections of the planes and the axis : and H any point in the plane EF. The centripetal force of the point H upon the particle P, exerted in the direction of the line PH, is as the distance PH ; and according to the direction of the line PG, or towards the centre S, is as the length PG. Therefore, the force of all the points in the plane EF, that is, the force of the whole plane, by which the particle P is attracted towards the centre S, is as the distance PG multi plied by the number of those points; that is, as the solid which is contained under that plane EF and the distance PG. And, in like manner, the force of
the plane c f, with which the particle P is attracted towards the centre S, is as that plane multiplied into its distance Pg, or as the equal plane EF multiplied into that distance Pg : and the sum of the forces of both planes as the plane EF multiplied into the sum of the distances PG+ P g; that is, as'that plane mul tiplied into double the distance PS of the centre and the particle ; that is, as double the plane EF plied into the distance PS ; or as the sum of the equal planes EF+ef multiplied into the same distance. And, by a like reasoning, the forces of all.the planes in the whole sphere, equally distant on each side from the centre of the sphere, are as the sum of the planes multiplied into the distance PS ; that is, as the whole sphere and as the distance PS jointly.
Case 2. Let the particle P now attract the sphere AEBF. And, by the same reasoning, it will be proved, that the force, with which that sphere is at tracted, is as the distance PS.
Case 3. Let another sphere be now composed of innumerable particles P; and, since the force, with which each particle is attracted, is as the distance of the particle from the centre of the first sphere, and as the same sphere jointly ; and therefore is the same, as if the whole proceeded from one particle in the centre of the sphere ; the whole force, with which all the particles in the second sphere are attracted, that is, with which that whole sphere is attracted, will be the same, as if that sphere was attracted by a force pro ceeding from one particle in the centre of the first sphere ; and therefore is proportional to the distance between the centres of the spheres.
Case 4. Let the spheres attract each other mu tually, and the force being doubled will preserve the former proportion.
Case 5. Let the particle p be now placed within the sphere AEBF ; and, since the force of the plane of upon the particle is as the solid contained under that plane and the distance p g; and the contrary force of the plane EF as the solid contained under that plane and the distance p G ; the force compound ed of both will be as the difference of the solids; that is, as the sum of the equal planes multiplied into half the difference of the distances ; that is, as that sum multiplied into p S, the distance of the particle from the centre of the sphere. And, by a like reasoning, the attraction of all the planes EF, cf in the whole sphere, that is, the attraction of the whole sphere is jointly as the sum of all the planes, or as the whole sphere, and as pS the distance of the particle from the centre of the sphere Case 6. And, if a new sphere be composed of in numerable particles p, placed within the former sphere