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We can again write down the system of equations which indicates 
the variability of the @’s and ?’s. This system has again as integrals; 
Ga ct, Cy i, a, 
—— — = constant, — + ——— = constant, +. —— = constant , 
n, Is Nm, AH, nN, n,n, 
a, + &, + A, = constant. 
Let us again put 
a a. a 
1 5 2 a) 2 3 y € 
—=%*, —=(—C,)7, ———=(C, — A, 
De nN, nn, 
then the integral A = constant takes the form: 
WEG ONE == Ees p = o (6 — 8), 
where / represents a constant and 
Q 
OF ee 
i Vn,n,(n, nh 
We now see easily in what way the coordinates are to be found 
as functions of time. 
§ 16. A survey of the general and special cases which can appear, 
as well as an insight into the manner in which the transition takes 
place on one hand to the case of the pure relation, on the other 
hand to the general case, where no relation exists, is to be obtained 
by representing the relation between 5 and g on polar coordinates, 
¢ being the radius vector, p the polar angle. In 
VES CONC ZS) cos p = o' (k—S) 
/ 
we may represent @’ as positive; for the curves for negative values 
N 
. / id . . . . Tv 
of 9’ are the mirror images with respect to the right line g = — 
of the curves for positive values of 9’. 
We give to C, and C, constant values and we find for a certain 
value of 9’ the forms of the curves satisfying the different possible 
values of /. We then see how this system of curves varies when 9/ 
passes through all values from very little to very large. 
For every value of @ three cases can be distinguished : 
1. & > C,. As § remains smaller than C,, the second member, so 
also cos p‚ remains positive. Curves on the right of 0. 
2.h<C,. As § remains larger than C,, the second member, so 
also cos ¢, remains negative. Curves on the left of U. 
3. C,<k<<C;,. The second member, therefore also cos p‚ becomes 
zero for S= k. Curves which surround 0. 
The curves represented by the above relation lie therefore either 
entirely on one side of QO, or they surround 0. 
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