APPENDIX. 425 



one, as to two such points, either in the same or different 

 planes from the body acted on. This is the fundamental 

 point in considering disturbing forces when the centres are 

 not fixed, which makes the problem more complicated and 

 difficult. It is, however, sufficiently so even where the 

 centres are fixed. 



2. That the subject must have attracted his attention 

 there can be no doubt. He had gone so much into the 

 more difficult inquiries respecting disturbing forces that he 

 must have fully considered the somewhat simpler, what 

 may be termed the fundamental, case of fixed centres. In 

 deed, a paper communicated to the Royal Society in 1769 

 (Phil. Trans, p. 74.) contains a demonstration by W. 

 Jones, an intimate friend of Xewton, of a proposition on 

 this subject, which Machin had immediately after Sir 

 Isaac s death given to the translator of the Principia. 

 Machin had observed on the want of some investigation of 

 the motion of forces directed to two centres, as required 

 to explain the motions of planet and satellite, which gravi 

 tate to different centres, in a word the problem of the 

 Three Bodies. The proposition of Machin and Jones goes 

 but a very little way to supply the defect complained of. 

 It is confined to the case of the line joining the two centres 

 being in different planes from the line of projection ; it is 

 that the triangle formed by the radii vectores and the line 

 joining the two centres or fixed points, describes equal solids 

 in equal times round that line ; and the demonstration is 

 similar to that of the first proposition, of equal areas in 

 equal times when a single force is directed to one centre. 

 It seems reasonable to conclude, that Xewton had, upon 

 full consideration, found the full investigation of the subject 

 beyond the powers of the calculus as it then existed. It 

 is at least certain that, though he might have mastered 

 it, he never could have delivered his results synthetically 

 as in the Principia. 



3. The solutions on disturbing forces generally consider 

 one force as acting in the one direction, that of the radius 



