132 Central Forces. 
Arr. XUI.—On Central Forces; by Prof. Toeopore Strone. 
(Continued from p. 342, Vol. X XI.) 
Ir is evident by the equation eae, that the angle v® beeen 
the apsides is always possible and finite when m is any positive num- 
ber >0; but if n is any negative number v° is always impossible ; 
A 
also if n=0 or F=—;, the angle between the apsides is infinite ; 
Prin. sec. 9, B. I, prop. 45, cor. 1, 
Aen, lel = = A ); c=const. then v9 =180° wie 
180° Lent or by expounding r (or R,) by unity, 0 =180° 
i Aerd ; 
<7 = ; Prin. cor. 2, to the prop. cited above. Finally, the 
— 4c 
curves denoted by (19) can be constructed by an ellipse whose focus 
is at the centre of force, R’/= its semi-parameter, and R/(1—e)= 
= its perihelion distance. For let the perihelion distance make 
the angle v — 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 7’ makes the angle v with R, and as the perihe- 
lion distance makes the angle v—mv with R, .". r’ makes the angle 
y—(v—mv)=mv with the perihelion distance; hence by the prop- 
R’ 
1-Le cos. mv 
quantities of the order e?; but by (19) r=R/(1—e cos. mv) .” 
7’=r and the particle is at the extremity of 7” (in the ellipse,) which 
makes the angle » with R. Hence by supposing the ellipse to re- 
volve around the focus, so that r’=r 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 »—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 v, and 
that of the perihelion »—mv, or so that the angular motion of the 
particle is to that of the perihelion as 1 ; 1—m 
erty of the ellipse 7’= =R’(1—e cos. mv), neglecting 
