MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
4, r ) 
For the range of values for u 0 from n 0 = 0 to w 0 = F 66 , the value of n would have 
a range of values from 2 to 1 * 2 . Thus the form of solution is materially the same for 
all of these values of u 0 . It will be seen without difficulty that the solution of (128), 
in which u has not a constant value, but varies over the range from 0 to 1 • 6 6 as r 
varies, will be such that the graph expressing y as a function of r will present the 
same features as are common to the graphs given by equation (130) for ranges of n 
from 2 to 1 * 2 . 
Now the value of y given by equation (130) is positive for all values of r, hence we 
infer that the solution of (128) is such that y is positive for all values of r. We 
therefore have, for all values of r, 
cr 3 = 1 ~ 3 + a positive quantity, 
so that <r 3 is positive for all values of r. 
§ 42. We therefore see that the initial motion, in which u and A are each 
proportional to the first harmonic, will first break down owing to the introduction of 
terms Involving the second harmonic. The sign of these terms is such that there is, 
in all the shells of which the nebula is composed, a diminution of density in the 
equatorial regions, and a condensation at both poles, which must be added to that 
given by the terms involving the first harmonic. 
The nature of this motion will become clearer upon a reference to fig. 3 . This 
figure consists of the four curves* 
r = cIq r = a 0 + a 1 P 1 
r = a 0 -f- a u P l fi- u 3 P 3 r — a 0 -f- cd n P 1 fi- a 3 P 3 , 
and these may be supposed to represent curves of equal density in the three stages. 
It is easy to see that of the pear-shaped surfaces of equal density, the equations of 
which contain the two first harmonics, some will be turned in one direction, and some 
in the other. For if they were all turned in the same direction the centre of gravity 
could no longer remain at the centre of co-ordinates. Thus, if the narrow ends of 
these pear-shaped figures point in one direction at infinity, we must, as we go 
inwards, come to a place at which they have the transition shape, namely, ellipsoids 
of revolution, and after this they will point in the opposite direction. 
It appears, therefore, that the initial motion is such as to suggest the ultimate 
division of the nebula into two parts, this division being effected by the outer layers 
condensing about one radius of the nebula, so as to leave room for the ejection of a 
* The particular values for which the curves are drawn are in the ratio a 0 = 11, a\ — 2, «xi = 5, a 2 = 2, 
a \i = 7, a '.2 = 4. Thus the equation of the last curves are in polar co-ordinates, 
r = (10 + 5 cos 9 + 3 eos 2 6), r = — (9 + 7 cos 6 + 6 cos 2 6). 
