250, 25l] 
Rotational Steady Motion 
281 
spherical. For such a star f(vr) must be zero, and the most general solution 
possible is that given by equation (2509). To a first approximation p 
may be replaced by p 0 , and so is a function of r only, so that cj> 2 is a 
function of r only. The circumstance that 0 did not enter into the equation 
of motion (247*6) has already shewn that if <j> is initially a function of r only, 
neither viscosity nor the transmission of radiation can destroy this property. 
Our present discussion has further shewn that for equilibrium (¡> must be a 
function of r only, at least so long as terms after the first on the right-hand 
of equation (250-8) are neglected. To this approximation, the rotation is by 
spherical shells, each spherical shell rotating as a rigid body with its own 
angular velocity, which in the outer layers of the star varies inversely as the 
square of the radius of the shell. 
251. To pass to a second approximation, we must take into account the 
second term on the right of equation (250'8). The additional terms required 
in the solution are seen to be proportional to sin 2 6 and cos 2 6, multiplied by 
functions of r only. Thus to a second approximation, the rotation in any sphere 
of given radius r is of the form 
(f> = A + BP 2 (cos 6) (251-1), 
where A and B are constants, or, in terms of the latitude X, 
(¡> = a — b sin 2 X (251-2), 
where b is of the same sign as B. To a still further approximation, we 
must introduce into </> a small term proportional to the fourth zonal harmonic 
P 4 (cos 6) giving a law of rotation 
(f) = a — b sin 2 X — c sin 4 X (2513). 
The physical explanation of these laws is that the more rapidly rotating 
layers in the inner regions of a star are of distinctly spheroidal shape, whereas 
the less rapidly rotating layers outside are nearly spherical. The inner spher 
oidal layers consequently approach nearest to a star’s surface at its equator, 
and their drag results in an equatorial acceleration, particles near the star’s 
equator revolving more rapidly than those near its poles. 
The mechanism can be simply illustrated by considering a star with a 
liquid centre of uniform density, built according to the extended Roche’s 
model. Let the liquid core rotate with a constant angular velocity w, and let 
its surface be 
x 1 + v 2 z 2 , cos 2 X sin 2 X 1 
a 2 6 2 ’ u 2 6 2 r 2 
If the angular velocity varies as 1/r 2 throughout the atmosphere which 
surrounds this core, the speed of rotation at the point R, X on the boundary 
of the star is 
• cor 2 g) a 2 
w = h 2 
i - 
l-e 2 
sin 2 X + 
e 4 
(1 - ej 
sin 4 X — ... 
.(251*4), 
18