86 
mi. J. H. JEANS ON THE EQUILIBRIUM 
at once to the equations deteriniuiiig the general configuration of this series, 
therefore replace equation (69) l)y 
l>7 = «^ + t(f^ + T) + 2 : 
/>,= X 
NT 
> 1=1 
We 
(70). 
This will be assumed to ]:)e the general form of the surface in the linear series 
now under discussion, the (juantity 6 l^eing a parameter wliich vanishes at the point 
of bifurcation. 
The equation expressing explicitly the solution of (70) may be supposed to be 
f = (1 - m, + Ifi + + . . .) . . . . (71). 
in which is tlie value of ^ when ^ = 0 , and tlierefore satisfies 
= «“ + I -^t) .(i"-), 
and i''^ series of ascending and descending powers of 17 , say 
6 = «o + “T + • • • -f +.(i^S). 
If we calculate the value of from (71), we find 
V; = — Trpfry + f7rp ( 1 
w \ 
u„ + .p 1 - t0-\_ 
(7-1), 
iTTp: “ ■ \ -'^P! 
where, if is given l)y (73), tlie value of U,, is 
Ib = Cl + «o (f + '1?) + (P + ■>?") +.(i*^)- 
Using the value of V,- given by (74), the equation to be satisfied at the surface is 
— ^77 (vrp — Twb + l-TTp 1 
\ 
Uq + vrp { 1 
\ 
O)- 
iirp ' 
const.. 
or, dividing throughout by irp — ^co'^, 
$r) = + terms independent of 6 .(~C). 
Ecpiation (76) must l)e identical with (70), the riglit-hand members of both being 
spherical liarmonics, and lienee W 7 e must have 
U, = ,Co + s” .C, (f' + V ’% 
n = l 
and therefore, by ecpiation (75), for all positive values of 
Instead of lining given by ecpiation (73), the value of may now be supposed to be 
given by 
$s — d-'i + “b 3,.Cbi7'' -!-•••+ ^ " "b 
(77). 
If we introduce the limitation that the curve is to remain of constant area, we must 
put c(_;^ = 0 . If we now replace cc_o, o _3 . . . by new unknoiviis /-•_!, ,vCL^, . . . , we 
can write e([uation (77) in the symmetrical form 
.(^ 8 ), 
in which we know that must ultimately be ecpial to zero, in order that the centre 
of gravity may coincide with the origin (cf. equation 22 ). 
