276, 277] 
Disturbances from Passing Stars 
309 
of encounters with other systems must be to spread these periods out over 
all possible lengths of period up to P = oo , the periods being fairly uniformly 
distributed except that a certain preponderance of short periods remains. 
277. The whole kinetic energy of a binary system can be expressed in 
the form 
MM' 
+ M') (w 2 + v* + O + \ uTTM'+ rW + r 2 sin 2 0</> 2 ) (277-1) 
or 
M+M' 
MM' 
i (M + M‘) c’ + i (r* + T»). 
In the final steady state, the average value of each of the squared terms 
in formula (277-1) must be the same, namely, 1/277, so that the average value 
of t 2 will be 
M_L M' 
(277-2). 
_ 2 _M + M' 
T ~HMM' 
ï)’(l_ e >)i. 
Since t = rd, the mean value of r 2 taken through the whole description of 
a single orbit is 
_s_ _1 [ p r2dt _ 1 _ » A . 7 /, 27 rh /27 
P) 0 P‘ 
For any value of P, formula (273*6) has shewn that the law of distribution 
of values of e is 2 ede. On averaging over all values of e, the mean value of 
t 2 for all orbits of given period P is 
2 /27 rfj,) 
Corresponding to the value of t 2 given by equation (277*2), the period is 
found to be 
2tt y(HMMf 
P = 
.(277-3). 
(M + Mf 
When a binary system has this period, the tangential velocity t has pre 
cisely the value appropriate to the final steady state, so that on the average 
encounters with other stars tend neither to increase nor to decrease this 
tangential velocity, and so tend on the average to leave the period as it is. 
Suppose, however, that a binary system has a period far greater than this. 
The tangential velocity t then has a value far below the average steady-state 
value given by equation (27 7-2), so that encounters with other stars are likely 
on the whole to increase this value and so to lessen the period. Similarly 
if a binary system has a period less than that given by equation (277*3), 
encounters with other systems are likely to increase the period. The law of 
distribution (276'3) may in a sense be regarded as a distribution ranged about 
the period given by equation (277*3) as median, but with infinite dispersion. 
On inserting the value H = 2 x 10 -47 already found, and measuring M