340 Central Forces. 
2c’? cot. *L : : 
——~“ as a quantity of the second order of minuteness, (be- 
cause ae described curve differs very little from a circle,) I have 
foro Ndr? 
Sat — a a Io? ie?) 3 by comparing this value of F° with its assumed 
2dr 
: dr? 
value there results the equation c’? d{ sa) =r(e’? —or), (15). Sub- 
2dr 
aoe % Z - dz? 
stituting R-+-a for r, in (15), and reducing it becomes d(53)= 
Qda 
xR oR vy? 
(@-+2)x (1-53) - (R+2) x <=> butl— 7g =I ye =a Mery 
yr 
small quantity, .". by neglecting quantities of the order x1 “yi 
ere , \ a/R. w7’R 
and quantities involving x7, also putting = + oR: (which may 
dx 
be done by neglecting quantities of the order x? ;) Ihave alae ie)* 
Qde 
dx? 
e’—m?x, multiplying by 2dz, and taking the integral Ie? =2e'a—m?2", 
dx 
(16), which needs no correction, since >- ae’ and a are each=0 at the 
“— e’3 
origin. Put 2e’x—m? ‘tm, sin.?9, then n= Ul — cos.) hence 
d 
and by (16) dy=— whose integral (since v and 9 are each=0 at 
' f 
the origin, where — gives 9=mv .". =~; (1 - cos. mv)s and 
r=R4+e=R+— —> cos. mv, (17). Let P= the semi-circum- 
ference of a circle, oi os l;) a it is evident by (17) that’ -s a 
maximum when mv=P, or v=—, F (18), at which time r=R+ 92? 
it is also evident that this value of 7 cuts the curve at right angles, 
and that » as given by (18), is the angle included by the greatest and 
least distances, or (as it is usually called,) the angle between the 4p- 
sides ; it is also evident that the particle after arriving at the greatest 
