EQUILIBRIUM OF ROTATING MASSES OF FLUID. 411 
omitted, since they are subsequently annulled by a proper choice of the angular 
velocity. 
From (22—i.) we have 
47ra 3 a 4 cttA 3 
"I” r ~3 ~c~ 
3 
2 
i 
r) r 5 
The last term in the development to the 7th order is that involving w Q . Then it is 
clear that we require h 2 correct to the 4th order, h ?j to the 3rd, and so on. But (25) 
shows us that the K s are equal to unity to the 4th order inclusive. Hence, in the 
above, all the h’s may be treated as unity. 
Again (22-ii.) when written in reference to the origin o affords other terms, in 
which all those included under 5A are of the 8th and higher orders, and negligible ; 
and the rest (with omission of the first harmonic term) gives 
47 rA*\-fa\ z Wz , 
"3tLw ^ + 
Thus this first part of the potential is, to the 7th order inclusive, 
47r« 2 a AirA 2, k - 6 /aV [ 3 /VA^ +1 
3 ‘r + , r 2 \c) [2 (/c — 1) [rj + 
Next, from the expression for n in § 3, we have a term in the potential due to 
rotation + ^co 2 r 2 . The remaining terms due to rotation will be taken up later. 
From (7l) we see that, to the 7th order inclusive, 
Hence cu 3 and e are of the 3rd order; and from (48) and (64) it follows that the 
factors by which the V s and ni s are derived from the X’s and /x’s are of the 5th order. 
And, since the X’s and p’s only differ from unity in terms of the 5th order, it follows 
that the l ’s and m’s are of the 5th order. Then (41) and (56) show us that all the 
terms in l and m are negligible. 
The first set of terms due to rotation and to the corresponding deformation are 
given in (39) and (43), and together contribute 
1 9 0 
ig-orcr 
+ U 
v h 
0 
r A 
The second set of terms due to rotation, and to the corresponding deformation, are 
given in (54) and (58), and together contribute 
1 2^,2 
6 w a 
Cl 
- + 7 
A 2' 
2 
3 G 2 
