82 
mi. J. H. JEANS ON THE EQUILIBRIUM 
The Series of Elliptic Cylinders. 
§ 17. The stable linear series is that of order 2. The ijiitial deforniatioii is therefore 
elliptical, and the point of bifui’cation occurs for the value or = np. 
The equation to the surface is initially 
iv = cr + CQ if- + f) . (oo). 
In § 13 we found (equation (44)) as the corresponding value of V, 
V; = 7rp {C — ^7] hx f)] .(5G), 
in which a. is a loot of 
a = ( I -f c'd). 
and this is true however great cq may be. The values of yi, (0, 0, 0, . . .) can 
accordingly be written down at once. We have 
/o (0, rto, 0, 0, . . .) = Ta, 
and all the other functions vanish. 
Hence there is a general solution of equations (52) in which all the cp/s vanish 
except and 
Ta = (l — orj'lTTp) a .2 .(3^)> 
where a is given by equation (57). 
This is the linear series of which we are in search. It is obviously a series of 
elliptic cylinders, and corresponds to the series of Jacoliian ellipsoids in the three- 
dimensional problem. From equations (57) and (58) we have 
orj-np = I — ad .(5^)- 
We therefore see that as we move along this series the value of oj- continually 
decreases from -np to 0. The angular momentum, however, increases from a finite 
to an infinite value. 
The Remaining [Unstahle) Series. 
§ 18. Before searching for points of bifurcation on this series, let us briefly examine 
the series passing the other points of bifurcation on the circular series, these series 
being known to be all unstable. Near the point of bifurcation the form of the series 
of order n is 
= cd -f a„ + .(GO). 
In § 11 we have calculated the values of/i (O, 0, . . . 0, . . .) as far as a,f If 
we neglect ad, it appears that all these functions vanish except /,„ and that /„ is 
given by 
/„ (0, 0, . . . a,„ 0, . . .) = afi ^{n — 1) cr'^-Upd. 
The series is accordingly given by equation (GO), until ad become appreciable. 
