78 
MR. J. H. JEANS ON THE EQUILIBRIUM 
= a^~ 4- {e^ + r) + Sh„ + 77'*).(40), 
1 
and let the solution be, as far as first powers of h’s, 
^0 + 2 ^ 1 ..( 41 ), 
1 
where is a function of 77 , and is the solution when all the h’s vanish. 
Substitute solution (41) in equation (40), and equate coefficients of h„, and we obtain 
^,,77 = 2«3 + 77 ", or, solving for 
= + .(42). 
We can express £q in the form 
^0 = + /^/V +.( 43 )- 
Hence equation (42) becomes 
(1 + u’‘) 7]’^ + ^ /St?" “ + . . 
T? (1 — 2floa) — 2fq /37]~^ + . . . 
L+n ^.-1 + ( 
1 — 2 rt 3 « ' ' yi — (1 — '2a^cx.)-J 
Hence we find as the value of V/, 
V, = TTp {C — ^77 + -la (^- +'>7')} 
77 ” d- &c. 
+ .p 2 (f' + V) jrri;;;; v + Ui-2«,« + (i _ 2a,«f } w 
+ terms in h„+Q, &c. . . . 
I ■ 
The value of £0 given bj equation (43) is a solution of equation (39). Sub¬ 
stituting this value, and equating the coefficients of the two highest powers of 77 , we 
find as the equations determining a and / 8 , 
a =z ( 1 + a^), ^ = cr -{- rto (2a/3 + 1 ), 
equations winch will be required later. 
Rotating Liquid Cylinder. 
General Theory, 
§ 14. \¥e now pass to the main problem before us, and consider the equilibrium 
of a cylinder rotating with angular velocity w. 
The equation to the cylinder for a rotation equal to zero is 
^77 = cu.(45). 
When the rotation w is different from zero, we shall' suppose the equation to the 
surface referred to its axis of rotation as origin to become 
