OF ROTATING COMPRESSIBLE MASSES. 
where 
16 5 
This expresses that V 0 is equal to the potential of a homogeneous solid of density 
p ti minus (p u —a-) times the potential E 0 , this last term accordingly representing the 
effect of the falling off of density from the maximum value p^. 
Similarly equation (ll) becomes 
where 
V; = poVi(l) — (p 0 — <r) E„ 
f 2 .,dV 0 (q) , , f 1 2 dV { (q) 7 2 
Ei = lT w dq ' + \, q w dq ~ 
dq 2 
■ ( 20 ) 
■ ( 21 ) 
in which the limit of integration is the value of q at the internal point at which the 
potential is being evaluated. 
The value of V 0 (q), as given by formula (16), is a function of q 2 and A', the two 
being connected by equation (17). Thus we have 
dV„ (q) _ 8V U (q) 0V„(g) 8 a' 
dq 2 dq 2 0g 2 ’ 
and the last term vanishes since dY 0 (q)/d\' contains \q J f+ <p (q)~\ as a factor. Thus 
we have 
dV o (g) = oV 0 (g) 
dq 2 dq 2 
— irClbc ( 1 
Ja' ‘ 
o0(g) \ 
og 2 / A 
( 22 ) 
and the value of dY, (q)/dq 2 is given by the same formula with X' put equal to zero. 
Using these values we find 
dtp (q)\d\ 
~ddT ! ^- 
dq 2 + Trabc 
/ d<p (q) \d\ 
\ dq 2 J A _ 
dq 2 . 
(24) 
Both values of E may be regarded as given by a double integration in a plane in 
which A and q are rectangular co-ordinates. Let us first consider E 0 . 
In fig. 1, let PQ represent the curve whose equation is 
+ 
if 
<r + A b 2 + A 
+ 
c 2 +A 
+</>(?) = </ 
(25) 
VOL. CCXVIII.—A. 
Z 
