Proposition 84

Table of Contents

PROPOSITION 85. THEOREM

If a body projected in some way is attracted to several centres of force A, B, C (Fig. 61), of which the individual forces are proportional to the distances of the body from these, [p. 289] then the body moves in the same manner, as if it is attracted equally by the common centre of gravity O of the points A, B, C in the simple ratio of the distances.

DEMONSTRATION

With the forces of the centres put at A, B, and C, and with the forces which they exercise at unit distance put as α , β , γ respectively, let ac be the direction of the motion that the body has at M, and thus the tangent of the curve EMF described at M. But O is the centre of gravity of the bodies α , β , γ put at the points A, B, C, and at O it is understood that the force varies directly with the distance, which at unit distance attracts with a force equal to α + β + γ . With these in place, a body at M is attracted to A by the force AM .α , at B by the force BM .β and at C by the force CM .γ . Moreover with these forces acting together, it is requires to show that a force equal to OM .( α + β + γ ) is attracting the body towards O . In order that this can be shown, perpendiculars Aa, Bb, Cc and Oo are sent from the points A, B, C, and O to the tangent ac.

In this manner any force can be resolved into normal and tangential components, and the sum of the normals arising from the attractions towards A, B,C is equal to : α .Aa + β .Bb + γ .Cc , and the sum of the tangents is equal to : − α .Ma + β .Mb + γ .Mc . But since O is the centre of gravity of the bodies α , β , γ situated at A, B, C, it has been proven from statics that : and

From which is evident for the forces acting together that α .AM , β .BM ,γ .CM is equivalent to the force ( α + β + γ ).OM . Q.E.D. [p. 290]

Corollary 4

Therefore the body according to this hypothesis describes an ellipse, the centre of which is placed in the centre of gravity itself O (631). For all the forces have the same effect, that a single force placed at O and attracting in the direct ratio of the distances.

Corollary 5

However many there are centres of this kind of force attracting in the ratio of the distances, the body still always moves in an ellipse, and as if it is entirely attracted by a single force at the common centre of gravity.

Scholium 1

The demonstration again succeeds in an equal manner, if some number of centres of force are not put in the same plane, as is evident from the principles of statics. Hence it is understood that the body is nevertheless moving in the same place, even if the centres of force are scattered in the most diverse of planes.

Scholium 2

If the centres of forces are attracting in some other ratio besides the simple ratio of the distances, a reduction of this kind to a single central position of the forces cannot be had in a straightforward manner, and I can hardly calculate the motion of the body, nor indeed can hardly anything be determined about the motion. [p. 291] Therefore in these cases it is necessary to flee to approximations, which are set up in different ways according to the various conditions. And on this account, Newton was unable to determine the motion of the moon, which arises from two attractions, but truly this is by far the most outstanding nearest attempt. Moreover, concerning this it is necessary to give this problem the most singular consideration, and the inverse method has to be called upon, where the body is receding from a known curve that it describes, under the influence of attracting forces. On this account, we will explain in the following what aids can be put in place, when we are to investigate the force acting as the unknown in the inverse order.

Therefore as we are progressing through this discussion, which can be established in two ways. Firstly, for besides the curve described is taken as known we take the direction of the forces acting at individual points, and from these quantities the forces acting, and the motion of the body itself is found. In the other way, by considering the curve and the motion of the body on that curve is taken as given, from which it is required to extract the force acting.