132 Central Forces. 
Arr. XI1.—On Central Forces; by Prof. Turopore Strouse. 
(Continued from p. 342, Vol. X XI.) 
Tr is evident by the equation Pa that the angle v° between 
the apsides is always possible and finite when x is any positive num- 
ber >0; but if m is any negative number v° is always impossible; 
also if n=0 or F= = , the angle between the apsides is infinite ; 
Prin. sec. 9, B. I, ne 45, cor. 1. 
Again, let = a ="); c=const. then v°=180° Cd 
: is 
dhior 
s9on/ Lae 
/1=e 
1— 40° 
curves denoted by (19) can be constructed by an ellipse whose — 
is at the centre of force, R’= its semi-parameter, and R/(1—¢)= 
R= its perihelion distance. For let the perihelion distance ei 
the angle »— mv with the line R drawn from the centre of force to 
the place of the particle at the origin of the motion; also let 7” de- 
note the value of r between the centre of force and perimeter of the 
ellipse. Now since r’ makes the angle v with R, and as the perihe- 
lion distance makes the angle »—mv with R, .’. 7’ makes the angle 
v—(uv—mv)=mv with the ae distance ; hence by the prop- 
erty of the ellipse. 7’=——— 
=» or by expounding r (or R,) by unity, p= 180° 
to the prop. cited above. Finally, the 
Fie tae 5. pa —ecos.mv), neglecting 
quantities of the order e?; but by (19) r=R/(1—e cos. mv) .”. 
r’=r and the particle is at the extremity of r’ (in the ellipse,) which 
makes the angle v with R. Hence by supposing the ellipse to re- 
volve around the focus, so that 7’=7 always makes the angle v with 
the fixed line R, the particle will always be in the perimeter of the el- 
lipse, and the angular motion of the perihelion distance will be v— mv; 
-". we may suppose the curves denoted by (19) to be generated by 
the motion of the particle in a moveable ellipse ; whose plane revolves 
around the focus so that the angular motion of the particle is 7, and 
that of the perihelion v—my, or so that the angular motion of the 
particle is to that of the perihelion as 1 ; 1—m 
