196 
MR. J. H. JEANS ON THE CONFIGURATIONS 
whence the mean density, as far as e 2 , is found to be 
P = A>(1— h + f&f) = />o [1 — fe—T40 6 "-”-+A(y — 2) e a . . (164) 
From these equations we obtain 
= 0T8712 + 0'06827e + (0’01602+ 0‘07098 (y-2)) e 2 +. (165) 
27 rp 
Ratio of Centrifugal Force to Gravity. 
41. [§ 41 revised August 31, 1918].—The ratio of centrifugal force to gravity is 
interesting only when it becomes equal to unity. When this occurs a stream of 
matter is ejected at the points at which centrifugal force equals gravity. Centrifugal 
force will first become equal to gravity at points which are furthest from the axis of 
rotation, and these will be points on the equator of the rotating mass. Considering 
a point on the axis of x , the condition for centrifugal force to be equal to gravity is 
dY/ox + (Ax — 0 , 
or 
dil/dx — 0. 
Since, by equation (30), 12 = —- y - p y ~ 1 + a constant, this may equally well be 
y-! 
expressed in the form 
dp/dx = 0. 
The general value of p is given by equation (45). On the x-axis this reduces to 
P = p 0 (l—eF), 
ICO , rx 4 , 2ux 2 
where 
F = ^ + 
a 
L.r 4 2 px 2 
a 8 + « 4 
e 
a 12 + a s + a 2 
+. (166) 
The intercepts on the axis of x are determined by the condition p = <r, and so are 
given by F = 1. The solution of this equation is found to be 
x 2 , (L 2 p 
2 — 1_ e \~ + 2 I ~ € 
a \a a 
R r 2u _ /L : 2p \ f 2L ^ 2p 
a 6 a 4 a 2 \a 4 a 2 )\a i ' a 2 
+. (167) 
The points on the x-axis at which cp/dx = 0 are given by 8F/3x = 0, so that it 
appears that dp/dx will just vanish at the extremities of the x-axis if dF/dx vanishes 
for the value of x given in equation (167). 
The equation oF/dx = 0 becomes, on division by '2x/a 2 
1+. 
2L /x 2 \ 2 p 
2 
3R /x 4 \ 2r (x 2 \ 2u 
_a i \a 2 i a 2 _ 
+ e 
_ a 6 v« 4 / a 4 \a 2 / a 2 _ 
+ 
- 0, 
