EVOLUTION OF THE STELLAR SYSTEMS. 229 
thereby converted into heat, and lost by radiation into surrounding space; thus the total 
energy of the system must decrease with the time. Hence it follows that, however the 
system be started, the guiding point representing the configuration of the system must 
slide down a slope of the energy curve. In the accompanying illustration the curves are 
drawn for the value of /Z = 4. 
If the guiding point is set at @ it may move either of two ways: it may slide down 
the slope ac, im which case the stars fall together ; or it may slide down the long slope 
ab, in which ease the stars recede from each other under the influence of tidal friction. 
This latter case is the one of chief interest in respect to systems actually existing in 
space, and the several other ideal cases need not be discussed in this paper. The con- 
dition at ais dynamically unstable, and corresponds to that of the system at the instant 
when the stars are first separated. At this juncture they rotate as a rigid system, but as 
each is losing energy by radiation, the axial velocities will soon surpass the velocity of 
orbital motion, and then the tides will begin to lag, and the mutual reaction of the stars 
will drive them asunder. Thus the guiding point in general slides down the slope ad. 
This means that as the stars recede from each other, the period of revolution for a long 
time surpasses that of axial rotation, but that in time the two periods again become 
synchronous when the guiding point has reached the minimum of energy at 4, where the 
bodies once more revolve as if rigidly connected. 
The question now arises with respect to the changes of the eccentricity. The 
differential equation for the change of the eccentricity is shown to be 
ff IT 
; = HB 
8a np = 
(@  @) ge Ese 
uU— 
— arcan —— 
ical 1a) sone \ aidan debates (5) 
J 
| 
I 
’ ih eee —$—__——__— 
= i eapea 
Roe 
where B is an arbitrary constant; a, 0, a + (, are the roots of the biquadratic equation. 
a’ — He + 2=0. Equation (5) is illustrated in the lower part of the preceding 
figure, the origin being shifted downward to 0’ to prevent confusion of too many curves. 
in one diagram. Now as the guiding point on the energy curve slides down the slope 
ab, the eccentricity at first very slightly decreases, then increases slowly, finally much 
more rapidly, until a high maximum is reached, after which it again diminishes, owing 
to the libratory motion in the system. Thus it is clear that as the stars recede from 
each other, the orbit becomes highly eccentric, but will ultimately become circular when 
