24 
MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
so that the equation itself reduces to 
tanh {1 - /) log (R,/R„)} = t A ~ « .... (88). 
This equation expresses the relation which must exist between R T /R 0 and p — p' 
(or, what is the same thing, between Rj/Rq and u), in order that p — 0 may be a 
solution of the frequency equation. 
§ 23. We shall be able to interpret this equation most easily by adopting a 
graphical treatment. If we write 
x = i (d - /*T> Vi = ~ ~ 
2 (n + i) 
y 2 = ^ tanh | y/x log (Ri/R 0 ) K 
x + (n + i) 2 J ' x/jc 
then the equation can be written in the form 
V\ = y%- 
It will be noticed that y % remains real when x is negative, an equivalent expression 
for y 2 being 
Vz 
v' — 
- tan { .y/ — x log (Rj/Rq)}. 
The roots of equation (87) are now represented by the intersections of the graphs 
which are obtained by plotting out y 1 and y 2 as functions of x. These two graphs 
are given in figs. 1 and 2 respectively, the graphs being drawn separately for the 
s ake of clearness. The graph for y x is, of course, the same for all values of R 1( /R 0 ; 
that for can be varied so as to suit any value of R(/R 0 ky supposing it subjected 
