308 
§ 12. The Equations of Motion. Possibility of Multiple 
Orbits at Low Temperatures. 
Since with small values of z we may write according to (4): 
Fy + yer +y7,2° +... (easing), 
this may always be reduced to 
F=a, sing + a, sn 3p + a, sindp +...), 
after which integration is possible. But since the expansion into 
series is not practicable after all for large values of z, in the neigh- 
bourhood of + a or — a, it is better to retain only the 1 term in 
the original equation, or to write: 
F:y,0(1 + Aat + Aat H..), 
and take the averages with respect to the factor between parentheses. 
Then we get: 
BVR nm YE: 
in which y will be > 0,655 e°: a’, when / approaches 2a. As now 
Yar 
sl é : : N 4 5 AS nende 
sin? p is == 4, the same with sin‘ p will be = —, with sin*p 
a) Es 24 
0 
: 163.5 
is = 546" etc., F becomes for /— 2a according to (6): 
0,655 e* 
IN: zee © (1 + 1,04 + 1,074 ...), 
hence averaged many times greater than y, 2. Let us now, for the 
sake of orientation, integrate the simple equation 
dek Pien 
MOLE BALIE etter afst We 
dt? , ( ) 
m mm 
in which y for small values of # will approach y,, when / — 2a 
(solid bodies and liquids), whereas for large values of / (gases) y 
approaches 12a? ¢?: / (see above). 
Thus we find with p= tnt de 
NES 6 8 
a teel, asing —acos6), . . . « (8) 
so that duly u becomes=u, at t=O (when M passes through the 
neutral point QO). Repeated integration yields for the path passed 
over: 
th Ten 
c=(u4+ 520 cos) (oe) = (asin p—asin 9), = to) 
