324 
The Ages of the Stars 
[ch. XII 
7 M vdoidR 
forces which vary from point to point of the system, and so change the 
period and eccentricity of its orbit. 
The number of distant stars which lie within a small cone of solid angle 
dm and at a distance between R and R+ dR is vR 2 dRda), where v is the 
star-density in this element of volume. If these stars have an average mass 
M, their contribution to the force X is 
5 + ¿ [® ( 2x ° 2 ~ y* ~ + ^y x °y ° + + • • .1 (291-3). 
In this expression x 0 , y 0) z 0 are each comparable with R. In summing over 
stars at all distances, the various contributions to the first term have the 
same relative importance as in the integral JdR, so that the important con 
tributions come from distant stars. Thus the motion in space of the binary 
system, as of course of any other star or system, is determined mainly by the 
distant stars. 
In the next term, different distances have the same relative importance 
as the corresponding terms in the integral jdR/R. The important contri 
butions to this integral come from large and from small values of R, the 
intermediate values being relatively unimportant. 
Although the contributions from large and from small values of R 
appear to be of approximately equal importance, they are effective to very 
different degrees in altering the period and eccentricity of the orbit of the 
binary system. The forces arising from very distant stars alter but little 
during the description of a complete orbit by the binary system, whence it 
is readily seen that the effects they produce at different points of the orbit 
tend to neutralise one another, and their total effect is very slight. The 
forces from near stars do not neutralise themselves in any such way, with the 
result that near stars are mainly effective in altering the orbit of the 
binary. 
The same remains true when we pass to the consideration of terms 
beyond those written down in formula (291-3), and we reach the general 
conclusions that 
( 1 ) Distant stars are mainly effective in determining the motion in 
space of a binary system or of a single star. 
( 2 ) Near stars are mainly effective in producing changes in the period 
and eccentricity of a binary system. 
292. The period P and angular momentum h of an elliptic orbit are 
given in terms of the energy E of the orbit by the equations 
P- 2?p y ( mm ' Ÿ. i ymm' ( mm' }*, * 
(m + mf V- 2 E) ' (m + {- 2 E) ^ > ‘ 
(m + my