Review of the Princvpia of Newton. 315 



tance, whence the formula for the centripetal or centrifugal 



v 2 1 



force F : -^-, becomes fr- 3 . 



As we have before observed, the various and copious re- 

 sults of our author's great problem cannot be fully exhibited 

 in our review. They have been spun out by his successors 

 into volumes, and may be found in the writings of Keil, Eer- 

 nouilli, Simpson, Maclaurin, Dawson, Matthew Stewart, and 

 others. The last writer has afforded some very elegant geo- 

 metrical propositions, illustrative of our author's work, on 

 this and other propositions of the Principia. There is, how- 

 ever, one corollary of too much importance to be passed over, 

 especially as it relates to the subject of the subsequent sec- 

 tion, viz. the motion of the apsides. The apsis of a trajecto- 

 ry, or orbit, is that point where the curve is perpendicular to 

 the radius vector, or where the paracentric motion ceases ; 

 whenever that is the case, the motion in the curve, and that 

 along the perpendicular to the radius vector, which constitutes 

 the generated element of the area, are equal. This determines 

 the apsis of the curve, which, however, in all cases may not 

 obtain, or if it should, it may be only for a determinate num- 

 ber of revolutions, and after that, the body may go off ad in- 

 finitum, or be urged to the center. 



Our great author, in order to show the motion of the ap- 

 sides in cases most applicable to the motion of the planets, 

 has devoted a whole section of his work to the investigation 

 of this subject, of which we shall endeavor to give some ac- 

 count. 



When the centripetal force is accurately in the inverse du- 

 plicate ratio of the distance, the revolving body may describe 

 either a circle about its center, an ellipse, or other conic sec- 

 tion about its focus. If the force be as the distance direct- 

 ly, it will describe either a circle, or an ellipse about its cen- 

 ter. These are the only laws of force, by which a body can 

 describe an ellipse accurately, or so that the curve should 

 perpetually return into itself, without any variation of the 

 similar and homologous parts of its orbit, or without any va- 

 riation of the position in space of that distinguished point 

 denominated the apsis. This point under the influence of 

 such forces must be fixed. If now we suppose the force to 

 act by laws differing from these, it is evident that the trajec- 

 tory cannot be a conic section, but some other curve, the ap- 

 sis of which must be a different point from that of a conic 



