VOL. LIX.] PHILOSOPHICAL TRANSACTIONS. 6O9 



cannot hold, whenever the body has a gravity or force to any other than one and 

 the same point, further says, " there seems to be wanting some such law as I 

 have here laid down, that may serve to explain the motions of the moon and 

 satellites, which have a gravity towards two different centres." 



About the year 1742, discoursing with that eminent mathematician, the late 

 William Jones, Esq. f. r. s. on the above-mentioned law, he showed me its 

 demonstration, and permitted me to take a copy of it; and which I conceive to 

 be highly worth preserving. 



Prop. — If a body p, fig. 1, pi. 17, projected in a given direction, be con- 

 stantly drawn towards two fixed points, s and t, which are not both in the same 

 plane with the direction, the triangle spt, fonned by right lines drawn from the 

 body p to those fixed points s and t, shall describe equal solids stpp', stp'p* in 

 equal times, about the right line st joining the said points. 



For, suppose a body projected in the direction pp', fig. 2, and acted on by two 

 centripetal forces towards the fixed points s and t: the angles p'ps, p'pt lying 

 in different planes. Let the time be divided into equal moments. In the first 

 moment, suppose the body, by its given force, should move along the line pp': 

 and in the second moment, if no new force was added, it should continue to move 

 in the same right along v'p ^ pp'; but when the body has come to p', suppose 

 it acted on by the two centripetal forces, in the directions p't,p's ; and let those 

 forces be in proportion to that in the direction pp', as the lines p't, p's to the line 

 v'p = p'p. With these three right lines p'p, v't, p's, complete the parallelopipid 

 p'p*; and the body in p', being acted on by these three forces, in the directions 

 r'p, p't, p's, which forces being as these three lines, shall move along the diagonal 

 of the parallelopipid made by these three lines; so that, in the second moment 

 of time, the body, instead of moving from p' to p, shall move from p' to p". 



Draw the lines sp',sp''' and tp',tp", as also sp,Tp. Now, the solid stpp' 

 = solid STp'p-, for they stand on equal bases tp'pjTp'/j, and have one common 

 vertex s, or their common altitude is the perpendicular drawn from s to the plane 

 PT/>. And the solid stp'p" = solid stp'/>; for they stand on the same base stp' 

 and He between the same parallel planes pt",st. Therefore the solid stpp' = 

 solid STp'p*. 



In like manner, in the third moment of time, the body at ?' being acted on 

 by three forces, in the directions p'p", p"8, p"t, shall move along the line p'p'", 

 so as to make the solid stp "p"' ^ solid stp'p"; and so in all succeeding equal 

 moments of time, the triangle formed by right lines drawn from the body to 

 the two fixed points st, shall constantly describe little solids, each equal to the 

 solid stpp'. Therefore the moments of the solids being proportional to the 

 moments of the time in which they are described; the solid itself is pro- 

 portional to the time in which it is described, a. b. d. 



VOL. XII. 4 I 



