EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
383 
These expressions in (14) are obviously solid tesseral harmonics. 
g2 JJ/, 
The transference formula required is for • 
By the formula (4) we have 
c?-*W i_ a _ 1 k_ + 7 — 2! w Jc . 
' Ji^-s ~ c, = 0 i — 21 k\ > ; 
operating on both sides by S 3 , and applying (14), we have 
c’~ 8 o 3 t| 7 _ 1 f — 2! 8 3 w /f 
i2 2i+1 i + 2 “ c * = „ i ~ 2! A! c* ' 
(15) 
Now the general formula (6) for the zonal harmonic shows us that d^wfldp 3 is zero 
when & = 0, 1, 2, 3, and hence S 2 m>* vanishes for the same values of k. Thus the 
summation in (15) is from k — co to k = 4, or, if we write k 2 for k, from co to 2. 
Hence (15) gives 
d Oo TT/* _ 1 A,” fl_ + d /«Y Bhvt +2 
A 2i + 1 “ i + 2 ~ c ,r 2 i - 2! * + 2! \c) a k 
This is the second transference formula required. 
We observe that the transference of a negative zonal harmonic gives us positive 
zonals, and that tesseral harmonics of the type S 3 W i+ JR~ lJrl give us harmonics of the 
type S 2 wt+ 2 - 
§ 2. The Mutual Influence of two Spheres of Fluid without Rotation. 
Imagine two approximately spherical masses of fluid of unit density, with their 
centres at the origins o and 0 respectively, and with mean radii a and A respectively. 
We shall find that each exercises on the other certain forces, one part of which has 
a solid zonal harmonic of the first degree as potential. This part of the force must 
remain essentially unbalanced in the supposed system, but we shall see hereafter that 
it is balanced by the rotation to be afterwards imposed on the system. 
Meanwhile it will be supposed that it is annulled in some way, and we shall content 
ourselves with finding the mutual influence of the spheroids, and the outstanding 
term of the first degree of harmonics. 
Let us assume that the equations, referred to our two origins, of the surfaces of the 
two spheroids, when they mutually perturb one another, are 
r 
a 
R 
A 
1 + 
1 + 
■^-K+l/aV 1 1 
aj i = 2 2i — 2 \c 
a \ 3 1 r. 00 2% + 1 (A\ i +1 
y 
i = 2 
2,-2 7 I 
(17) 
The It s and H’ s are unknown coefficients, to be determined. 
