Central Forces. 67 



4r'rcos.-u.' . [r' -^r)- —c^ =47nn=4r'r cos.^u= by 



cos.s-cos.^^ 



p' I v' v\ 



(8) and (12); or -^ (^l+tan.^- tan.^j ='v/mn; also w+n — 



2^mn—Wm-^^nY=r'-^r~2Vmn—^ [^ 1 +tan.==2 + 1 + 



v\ I v' I v\ p' I v' v\^ .— /_. 



tan.2 2J —V' ^l+tan.^tan.^ j =2 ^tan.^-tan.^j ; or(v m-V w) 



. /f' p' I v' v\ 



XV -2-=-2"( tan.^ — tan.^ |. By substituting the values thus 



/ X)^ 3 3 



found in terms of m and n, in (14), it becomes 'v 7^x(m^ — n"^) 



=3(A' — A) ; hence by restoring the values of A', A, m^ n, and put- 



3 3 



ting y=gl have i'-^=^ '-^^^ (15); (See 



Mec. Anal. Vol. II, p. 31.) 



Suppose eZ. 1, then the conic section is an ellipse; «= its semi 



transverse axis, and e= its focal distance -^a. (6) is easily chang- 



e la(o . \ . , 



edtoTC^— -( — — asm. (pl> or smce w, e, a, are the same at all 



. <^?* . 

 points of the curve, t is as — — a sin. 9 ; which indicates Newton's 



SO 

 construction, (Prin. Vol. I, Sec. VI, prop. 31,) for ^='Tr\ in his fig- 



a AO- a-^ 



ure, rt=AO, .•.~=-^^= hisOG, and 9= angleAOQ .•. — = OG 



X (p = arc OF, also a sin. 9= AO X sin. AOQ= sin. AQ (to rad. AO ;) 

 .-.— -asin.(p = GF— sin.AQ^GK, or i is as GK. Put in (6) 



<p=9'-fj:(a?=a small arc) then n^— 9'+a; — esin. {cp'-{-x)=cp'-\-x — 



nt — ^^-{-e sin. cp' 

 e sin. cp' —xe cos. cp'r.x= — :j — — ■ . , — neglecting terms which in- 



volve x^,x^, he. ; or if a?°, nP, 9'°, R°, denote the degrees in x, nt, 



o', and an arc of a circle equal to its radius, x°=^ ; 



^' ^ 1—e COS. cp 



(16). (16) agrees with the first method of approximation given in 



the scholium to the 31st proposition ; for nt° = 'N, q;''°=AOQ, eR^ = 



