Central Forces. 67 
*C0S.7 4° 
4r’r cos.?u.*. (1 +-r)? —c? =4mn=4r'r cos.2u= . by 
oe — 
Co0s.2= cos.? = 
i. og 
(8) and (12); or = (1 +tan Fey 5) =Vmn; also mtn— 
2V mn. =(V im — Vn)? =r! +r 2 mn = Fi 1 + tan. = 5+ 1 +- 
v aot 
tan.? 3) -p (11a. 5 tan. 3) af (tan. +g tan. 5) 3 or (Wm-Vn) 
ae y= (tan —tan.5). By substituting the values thus 
af 
found in terms of m and n, in (14), itbecomes Ve v4 (m* —n?) 
ued —A); hence by restoring the values of A’, ™ m,n, and put- 
+trte)?—(r+r—- 
ting oe 1 have begak sf { ? (15); (See 
Mec, Anal. Vol. Il, p. 31.) 
Suppose eZ 1, then the conic section is an ellipse ; a= its semi 
transverse axis, aod e= its focal distance ~-a. (6) is easily chang- 
e (ap : 
ed to aime (2 —asin. 9)» or since n, €, a, are the same at all 
a eee fone ; 
points of the curve, f is as ] asin. g 5 which indicates Newton’s 
eee Lee 
construction, at Vol. I, Sec. VI, prop. 31,) for e=y6 in his. fig- 
ure, a=AO, . a his OG, and o= angleAOQ ...— = OG 
Xo=are GF, also asin.g=AO Xsin. AOQ= sin. AQ (to rad. AO;) 
a 
a sin. >=GF — sin. AQ=GK, ortis asGK. Put in (6) 
?=9'+ 2(@=a small arc) then nt=9/+2—esin. (p’+-2)=9’+a— 
nt —o’+-e sin. of 
1—ecos. 9’ 
volve “7,27, &c.; or if 2°, nt°, 0°, R°, denote the degrees in z, nt, 
; nt? —g/°-+- eR°sin. of 
9; and an are of a circle equal to its radius, 2° = Peasy 
(16). (16) agrees with the first method of approximation given in 
the scholium to the 31st ; for nt°=N, o/°=AOQ, eR°= 
££ 
esin. 9 — xe cos. 9’. .2== neglecting terms which in- 
