25 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
to an appropriate uniform extension parallel to the axis of y, and contraction parallel 
to the axis of x 1 or vice versa. Similarly, different values of (n -fi \) can be 
represented by contraction and extension of the first graph. 
If we imagine these two graphs superposed, we see that there cannot, under any 
circumstances, be an intersection in the region in which x is positive, i.e. (equation 
(84)), for a value of u less than (n + |-) 3 . The lowest value of u for which an inter¬ 
section can possibly occur is u — 1, and this occurs only when H 1 /H 0 = co . As R L /R 0 
decreases from infinity downwards, the lowest value of u for which an intersection 
occurs will continually increase. Whatever the value of Rj/R 0 may be, there are 
always an infinite number of intersections in the region in which u > T (n + -f) 3 . 
The values of u found in this way determine the “ points of bifurcation ” on the 
linear series obtained by causing u to vary continuously. Thus we have seen that as 
u continually increases the first point of bifurcation of order n is reached when u has 
a value which is always greater than T (n + T) 3 . When Rj/R 0 is very large, the first 
point of bifurcation is of order n — 1, and its position is given by 
«=U.(89). 
§ 24. Let us, in future, confine our attention to the case in which R^Rq is very 
large. If we gradually remove the restriction that u is to be independent of r, the 
various vibrations of frequency p = 0 will vary in a continuous manner. Equation 
(55) remains unaltered in form, and, at infinity, it assumes the definite limiting form 
e^={»(«+l)-2«,}f. (90). 
where u„ is the limit (supposed definite) of u at infinity. It therefore appears that 
at infinity the solution for £ approximates asymptotically to that given by equation 
(82), if y , y! are now taken to he the roots of 
t (t — 1) = n{ii + 1) — 2u„ . . . . . . . (91), 
Equation (85) accordingly remains unaltered. Equation (81) takes a form which 
is no longer represented by equation (86), but which will impose some definite ratio 
upon E 1 and E 3 . It is therefore clear that when R x is very great, equation (85) can 
only he satisfied, at any rate so long as /x and // are real, by taking /x — /x' very 
small. Thus a point of bifurcation will again be given by /x — jx = 0, our previous 
investigation sufficing to show that this gives a genuine solution to our problem, and 
does not correspond to an irrelevant factor introduced in the transformation of our 
equations. This point of bifurcation is moreover the first one reached as u increases, 
since it is at the point at which /x, y! change from being real to being complex. 
VOL. CXCIX.—A. 
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