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On the ORBITS in which BODIES revolve, being aEled upon by 

 a CEN,TRIPETAL FORCE varying as any FUNCTION of 

 the DISTANCE, -when thofe ORBITS have TWO APSIDES. 

 By the Rev. J. BRINKLEY, J. M. ANDREJVS Frofejfor of 

 Jljlronomy in the Univerftty of Dublin. 



HE inveftigation of orbits defcribed by bodies adled upon by Read March 



T 



any centripetal force whatever is reduced by Sir Ifaac Newton to 

 the quadrature of curves (8 fe6t- lib. i. Princip.) The quadra- 

 ture of fuch curves as arife from the aj^plieation of his method can 

 only in few inftances be completely accomplifhed. A portion of 

 the area of any curve may be eafily found by a converging feries, 

 but not fo the whole area. To approximate to the whole area is 

 in moft cafes very difficult; and hitherto the orbits have been 

 inveftigated for very few laws of force. By the method 

 here propofed it is fhewn, that when the orbit has two 

 apfides, that whatever be the fundion of the diftance which ex- 

 prefTes the law of the centripetal force,^ the orbit rnay be deter- 

 mined by a feries of fines of multiple arcs converging by the 

 powers of the excentricity. From hence the angle between the 



D d 2 ', apfides 



9, IBOl. 



