1(58 
MR. J. H. JEANS ON TPIE VIBRATIONS AND 
Points of Bifurcation. 
IG. The interest of the question lies in the position of the points of bifurcation ; 
to find these we must put if ■= 0 in tlie frequency equation. The reason why it was 
not j)ossible to put jr = 0 at an earlier stage will l)e understood hv those who have 
read the former paper “ On the Stability of a Spherical Nebula.” In the present 
instance it is, perhaps, sufficient to say that putting p~ = 0 at an earlier stage would 
have led to an entirely misleading result. Upon putting jf = 0 in equations (50) 
and (51) we find that the two Ijrackets multiplying vanish, and we therefore see 
tluit must he treated as an infinite quantity of the order of l/p-. 
Expanding these brackets as far as jf, and then putting p- = 0, we find that the 
two equations l^ecome 
- Wfh - 0 
wliere 
(52), 
a’j = (?!I + 
T-n-g ( 2 /i. — 2 ) c 
V- 
dr 
X. = V, 
_ (/? — 1) (2/; + Ifi per I 2(2)1- — 1) 3/i + 1 
' ‘ ‘P'^p M + 1) (2)1 + “ 2{,i + 1)] 
y-z = 
per 
n 
2 ( 2 ) 1 + ])(2a + 3) 
The equation giving points of bifurcation is, of course, 
. (53), 
+ ‘Uffi = 0.(54) 
ihe values of and x.^ are found, after some simplification, to lie 
(« — 1) (2)1 + 1) d 
^ (ku) -f- 20 (/cu ) — na (/<«) 
2(n - 1) (2a +1) 
n 
+ 
■)IK 
{kC() 
del 
. . . (55), 
U — ^ + 1 fa) + 2u-0/f'J„+5 fa) — 2 (n -f l) {kci) . . ( 50 ). 
Now, it lias already 1jcen seen tliat 0 —-- (p. 104). If we substitute this value 
K- 
loi (.•, V 1 ite X lor ko, and sliiq)]ify equations (55) end (50)) as far as possible, we have 
J, ,, ( A -f J / X 2 {)i - 1 )( 3 « + 2 ) 
‘ + ;-r J.+i. (U- - -J.+Ua.’) • (5/), 
X a~^ _ N + , 2 (a + 2) 
XP - ^ -- - J„^,(.r). . 
(58), 
