340 Central Forces, 



2c' 2 cot. 2^|. . . - 

 1 as a quantity of the second order of minutenesi?, (be- 

 cause the described curve differs very litde from a circle,) I have 



c'2 c'^ idr^v 

 Y=^ — —ld ( TTJ ; by comparing^this value of F with its assumed 



2dr ■ . 



value there results the equation c'-^rr"^) =i'{c'- —(fir), (15). Sub- 



2dr 



fdx^\ 

 stituting R-fa; for r, in (15), and reducing it becomes c? -r~^ j = 



2dx 

 (R+^) X (^ I - -,TJ - (R+^) X ^^ 5 but 1 -^ = 1 ~yi- = a very 



small quantity, •'• by neglecting quantities of the order x\ 1 -y^ , 



a?/R a?:p'R 

 and quantities involving x^ , also putting ,^ = — t5~5 (which may 



■ • • idx"\ 



be done by neglecting quantities of the order x"^ ;) 1 have dy-r^ = 



2dx 



dx? 

 e'—rn-x, multiplying by 2dx, and taldng the integral -j-^ =^2e'x—'m^x-, 



dx 

 (16), which needs no correction, since -7-, and x are each=0 at the 



e'2 . e' 



origin. Put2e'a; — wz^a;^=-^ sin.-ffi, then «= — ^(1 — cos.®.) hence 



df) 

 and by (16) dv = — whose integral (since v and (p are each — at 



e' ' 



the origin, where x=^Q;) gives (p='my .'. a:= — ;(1 — cos. mv), and 



e' e' 

 r=R-l-a:=R+— 7 — — 7 cos. mu, (17). LetP=the semi-clrcum- 



ferense of a circle, (rad. = l,) then it is evident by (17) that r is a 



P 2e' 



maximum wnen mw=P, or v — —, (IS), at which time r=:R+-^ ; 



it is also evident that this value of r cuts the curve at right angles, 

 and that v as given by (18), is the angle included by the greatest and 

 least distances, or (as it is usually called,) the angle between the ap- 

 sides ; it is also evident that the particle after arriving at the greatest * 



.« 



