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ration of a centripetal force, varjnng in a compounded ratio of the 

 diftance, and confequently that the 2d Cor. of the 45 th Propofition 

 of the I ft Book of the Principia is erroneous. The common error 

 in thefolutions of Walmfly and Frifius is hereafter pointed out. 



The examples of the method here ufed are given in the 2d arid 

 3d propofitions. The ift propofition proves, that an orbit, hav- 

 ing two apfides, and defcribed by a body impelled by a force vary- 

 ing inverfely as the fquare of the diftance, is an ellipfe. This fo- 

 lution is probably more dired than any other of' the fame propofi- 

 tion, and has a connedion with what follows. The feries in the 

 2d and 3d propofitions are not continued beyond the 2d and 3d 

 powers of the excentricity. This was fufficient for the motion 

 of the apfides, and there can be no difficulty in-continuing the 

 terms. 



It feems hardly necefTary to obferve, that when the force varies 

 as any fundion of the diftance, an orbit can have only two diffe- 

 rent diftances of the apfides from the centre, becaufe it muft be 

 fimilar on each fide of the apfid. The law of the force alfo readily 

 fliews, v/hether any orbit defcribed by that' law can have two 

 apfides, by comparing it with the law of the centrifugal force, 

 which always varies in the inverfe triplicate ratio of the- diftance. 

 Therefore if the force varies in any dired ratio, or any inverfe 

 ratio lefs than the triplicate, the orbits defcribed may have two 



apfides. 



' Prop. 



