209 
So we have the following cases: 
A. a° 24. z lies between — 4 and 2; formula (33a) holds; 7 
varies between oo and «; the first value is attained for 
1 a7 B get 
f= fpf, = —>— log EEn en 
ghs aV3 WV 2d 
and the second for 
ESR VE 
PE — log mn ed 
rope Be Wda dà 
An infinitely long time is required to reach either position. 
B. a? <4; 2 between a and 3. Formula (33) must be applied; r 
varies between «a: (& + a’) and «; p then changes from oo to 
if V 2a? En aV3 
Ps = —— log > 
V 2a? + amal. V3 
The orbit comes from 7 —e and approaches in a spiral to circle 
Pi dU). 
C. 4 <a’? <3}; 2 between — + and a’. Formula (33a) now holds; 
r varies between o and ea: (4 + a’); p changes from /, to oo. The 
orbit comes from infinity and turns in a spiral round the circle 
7 —a:(4+ a’), which lies between circle 2« and circle a. 
D. a? <#; 2 between — 2a’ and a’. Formula (2a) must be 
applied; 7 varies between a: ($ — 2a’) and «:(4 +a’); p changes 
from 0 to oo. The orbit is a spiral, coming from circle a: (4 — 2a’), 
which may have any radius > 3a, and approaching in a infinite 
number of turnings to circle «:(% + a°), which lies between circle 
2a and circle 3a. 
12. Now we will suppose the roots @,,¢,,e, to be all different. 
As regards ‘these roots, we may then distinguish two main cases, 
viz. the case of three real roots and the case of one real and two 
conjugate complex roots. In the first case we put e, >e, >e,, in 
the second e, be the real root and the imaginary part of e, be 
positive. In either case we put, as usual, ¢,— Pw,, €, = Pw,, 
é, = Pwo, with ow, ==, Hw, (not — w, —a,). 
The three roots are real. The only valnes possible for ¢ s in equation 
(25) now are O and w, (or congruent values). In the first case z 
varies from oo to e, and from e, to oo, while p changes from O to 
2w, and from 2, to 4w,. One must, however, remember that, 
according to (27), z may not exceed the values — } and # (i.e. rr = oo 
and r= a), but must remain between them. So if e, >, it is 
impossible for 7s to be zero. If e, < %, 2 varies between e, and 3 
and so r between «/(4 + e,) and «. This case corresponds to 10 and 
14 
Proceedings Royal Acad Amsterdam. Vol. XIX. 
