( 625 ) 
Now put: 
/(0 = g«(lHE>-*\ 
then for the initial value of g we find /(g) > 0. For g = 0 and 
g=l we find /(g)< 0. Thus the equation /(g) = 0 has two roots 
between 0 and 1. 
So K* cannot have all values; the possible values of K* lie between 
two limits; in § 9 we shall revert to this and to the special cases, 
corresponding to the limiting values of K *. 
The roots between 0 and 1 which the equation 
g* (1 — g) —■ K* — 0 ....... (14) 
has in the general case will be called g x and g a , where we suppose 
5 s !>Si- The third root is negative, we call it —A. 
The differential relation between g and t may now be written: 
= - = .(15) 
t / <S,-5)(?-5.)(5+i) N 
So with the aid of elliptic functions g may be expressed in t. 
It changes periodically between the limits g, and g t . 
Now with the aid of (12) we can also calculate <p as function of 
t. And ft and & likewise, it being possible to write the last two 
equations of (9): 
d^_ d 2 R 0 K h 
dt~ 2 JV* ’g’ 
dp 2 _d,R 0 K h 
dt~ 4A* 1—g' 
So now x and y are also known as functions of t 1 ). 
In fig. 1 the relation (12) between g and <p is represented in 
polar coordinates, <p is taken as polar angle, —g as radius vector. 
The circle drawn has unity as radius. The curves change with the 
value of K. For K> 0 the curves lie to the right of the straight line 
<P = ~ , for JT<0 to the left of it; K=0 furnishes degeneration 
into the straight line <P = ^ the circle g = 0. By the maximal 
2 
positive and negative value of K(K—±- 1/3) the curve has 
contracted into an isolated point. The special cases of the motion 
belonging to K= 0 and to K= ± ^ |/3 will be discussed in $ 9. 
l ) These calculations will be found in my dissertation, which will appear before 
long. 
