66 Central Forces. 



a qcos.(v — 6) . ^ I ^\ 



COS. d COS. d ^ ' ^ 2 -^ \ 2/ 



V 



5 14-tan.--cos.^. , ,; t,\ 



orgtan.^= ^^^ =-^\^\^n.^^i^n.'^) , ovhy [2] 



3A 

 qia.vi.^=^, (11)- (11) agrees with Newton's construction, (Prin. 



A 



Vol. I, Sec. VI, prop. 30.) his M=^^^ and otan.5= his GH= 



zp 



3M. Let p= the semi circumference of a circle rad. being 1, then if 



p V p'^ 3A p' p- 



t;=-» tan.^ =1, and A becomes "o"' hence ^ : -^ : :A : ~7r~' 



3A 



which agrees with Newton s proportion m cor. 1. and ^-tan. ^=-^-, 



J.p 



Sc't d(q tan J) 3c' 3V ^^^ ^ ^ . ^ , 

 = 4^ S'^^s Jt "" ^"""8" (^"^ ^^^® velocity of the par- 



d{q tan J) 

 tide at the vertex ;) or -j: : \ ! ; 3 : 8, which is his propor- 

 tion in cor. 2. and his cor. 3 is evident by (10) which are assumed 

 on the supposition that a circle is described through the points A, S, 

 P, in his figure ; its centre H being at the intersection of GH and a 

 perpendicular erected at the middle of SP or r, its radius SH=R, 

 GS=q, HSG=^, ASP=r, .•.HSP=y-^. 



Let r, V, t, A, become ?-', v', t', A'; then by (8) r^= ■,= 



2cos.-^ 

 2 



|^l+tan.=2J (12); andby(9)'-^^3tan.2-+tan.3-j =3A' 



(13); by subtracting (9) from (13), resolving the remainder into 



V v' ^ ■ 



factors, and putting for l-j-tan.-^' l-j-tan.--^> their respective 



2r 2r' p' I v' 



equals — ' —7 as given by (8) and (12) ; there results-^ I tan.^- — 



tan.|) X (r'-l-r-^-|-(l+tan.^tan.^)) =3(A' -A) (14). Put^;'- 



'21 \ ' • ' 2 ^ ^ 2 2' 

 V=2u and let c= the chord connecting the extremities of r', r. and 

 r'-{-r-^c=^2Tn,r' •^r — c — 2n .' . r^-^r—m-^-n ; (by trig.) c^=r'^-\- 

 f' —2r'r cos.2u, 01 since cos. 2r<~2 cos.-m— 1, c= =(?-'4r)^ *- 



