his 



Central Forces. 

 SO 



is B, for '2=To (^^^ ^^^^ ^§') ' ■ ^^ ' ^^ •" : R^° • eR° which agrees 



R 



with his proportion for finding B ; also his L= — (R= his radius ;) 



R R ^ 



. ■ . L ; L — R cos. (p' : : — : — — R cos. 9' ; : 1 : 1 — e cos. (p' ; and by 



(16) 1 — e cos. (p' : 1 : : nP — 9'°-f eR° sin. 9' : .r°, which agrees 

 with his proportion for finding E, •'. his E=a?°. By adding x° to 

 9'° and repeating the operation with the corrected value of ip'^, 

 the second part of his process will be obtained, and so on ; ob- 

 serving that the successive corrections are to be applied according 

 to their algebraic signs ; and that the sign of e cos. 9' in the de- 

 nominator of (16) must be changed into -}- when 9^° is between 90° 

 and 270°, also the sign of fR°sin.9' must be changed into — when 

 9'° is between 180° and 360°, because cos. 9' is negative in the former 

 of these cases, and sin. 0' in the latter ; when 9 is found, v is easily 

 found also by (T). By changing the sign of e cos. v in the denominator 



a(l-e-) 



of (2) it becomes r=z (17) ; v being counted from the 



^ ^ l — e cos. V ^ -^ ^ 



aphelion. Put 2a — r=r', then r'= the distance of the particle, 



which is supposed to describe the ellipse, from the other focus ; let 



v^= the angle made by r' and the distance of that focus from the 



a(l-e2) ^ ^ 



nearer vertex, then r' = zr—, r (18) : and because r, r' make 



' 1-j-ecos. v' ^ •" ^ 



equal angles with the tangent at the place of the particle rdv=^r'dv', 



a-\-2aeco%.v'-]~ae^ 



OYr"dv=c'dt=rr'di/ (19). By (18) r=2a-r'= ~ ; » 



^ ^ .? V ^ 1-j-ecos. V 



and c'c?^ = '\/^p' xdt = 'v ag[i — e'') xdi ; by substituting these 

 values, and that of r' as given by (18), in (19) and reducing, I have 



(l-f2ecos. v'4-e'-')X'^l -e" Xdv', . /^ 



(1+ecos. v')^' a^ ' 



il-\-2ecos.v' -\-e" Qos.^v'\ _i 



ndtx [^ i+2ecos.t' + e^ ) x(I-e^') ^ or since cos.^z-'^ 



l-f-cos.2ir' 



~ by rejecting quantities of the order e*, e^, &c. I have 



dv'= ( 1-f-^cos. 2i'''+e3(cos. ?j''-cos.'y'cos.2t»^) 1 XnJ^;by known 



cos. 3d'-}- cos. v' 

 formula cos. «' cos. 2i/= ^- J hence and by neglect- 



