Finding the orbits from the focus given

PROPOSITION 88 THEOREM 45

If the attractive forces of the equal particles of any body be as the distance of the places from the particles, the force of the whole body will tend to its centre of gravity ; and will be the same with the force of a globe, consisting of similar and equal matter, and having its centre in the centre of gravity.

Let the particles A, B, of the body RSTV at tract any corpuscle Z with forces which, supposing the particles to be equal between themselves, are as the distances

Let the particles A, B, of the body RSTV attract any corpuscle Z with forces which, supposing the particles to be equal between themselves, are as the distances AZ, BZ; but, if they are supposed unequal, are as those particles and their distances AZ, BZ, conjunctly, or (if I may so speak) as those particles drawn into their distances AZ, BZ respectively.

Let those forces be expressed by the contents under A × AZ, and B × BZ. Join AB, and let it be cut in G, so that AG may be to BG as the particle B to the particle A; and G will be the common centre of gravity of the particles A and B.

The force A × AZ will (by Cor. 2, of the Laws) be resolved into the forces A × GZ and A × AG; and the force B × BZ into the forces B × GZ and B × BG.

The forces A × AG and B × BG, because A is proportional to B, and BG to AG, are equal, and therefore having contrary directions destroy one another.

There remain then the forces A × GZ and B × GZ.

These tend from Z towards the centre G, and compose the force A + B ¯ × GZ; that is, the same force as if the attractive particles A and B were placed in their common centre of gravity G, composing there a little globe.

By the same reasoning, if there be added a third particle C, and the force of it be compounded with the force … × GZ tending to the centre G, the force thence arising will tend to the common centre of gravity of that globe in G and of the particle C; that is, to the common centre of gravity of the three particles A, B, C; and will be the same as if that globe and the particle C were placed in that common centre composing a greater globe there; and so we may go on in infinitum.

Therefore the whole force of all the particles of any body whatever RSTV is the same as if that body, without removing its centre of gravity, were to put on the form of a globe. Q.E.D.

Corollary

Hence the motion of the attracted body Z will be the same as if the attracting body RSTV were sphærical; and therefore if that attracting body be either at rest, or proceed uniformly in a right line, the body attracted will move in an ellipsis having its centre in the centre of gravity of the attracting body.