390 
PROFESSOR. G. H. DARWIN ON FIGURES OF 
The last terms of (35) and (36) are constants, and the term in r 2 , and that in R- 
are symmetrical about each origin, and so the corresponding forces can produce no 
departure from sphericity in either mass ; thus these terms may be dropped. ISText 
we have in (35) and (36) the outstanding potentials — ardw x and — orD W x , which 
will be annulled by other similar terms, and so need not be considered now. We are 
left, therefore, with the terms in q 2 and w 2 , or in Q 2 and W 2 . The q. 2 is a sectorial 
harmonic, the w 2 a zonal, and it will be convenient to treat them separately. We 
shall begin with the zonal term. 
§ 4. Disturbance due to the Zonal Harmonic Rotational Term. 
The potential whose effects we are to consider is or ^co 2 W 2 , according to the 
origin which we are considering. 
If an isolated spheroid of fluid of unit density be rotating with angular velocity cc, 
the ellipticity of the spheroid is 15aj z /16n; therefore we put 
(37) 
Let us assume, for the equations to the two spheroids, 
r 
a 
R 
A 
, , t w % , (Ay { ^° 2i + 1 fa\ i + 1 wi 
1+ ^ + [a) i ? ! 2TT2W 
, , i W "- , /“Y'V 2i + 1 l A Y +1 r w > 
1 + 3£ ^+L) Li * 
(38) 
where li, L n are unknown coefficients which are to be determined. We now have to 
determine the potentials at any point of the inequalities (38) on the two spheroids. 
The potential of the inequality ■§•€ w 2 fr % in the first of (38) is 
7T a°. a 
2 1 . 
(39) 
The similar inequality in the second of (38) gives us 
in A 3 . A 2 . 
4t rA s 1 /A\\c 3 W 2 
3 c 5 e \ c ) D 
(40) 
The term in l k in the first of (38) gives us, as in § 2, 
47 rA 3 34 /«y /«y + 1 Wjc 
3c 2 k — 2 \c / -\r/ A 
( 41 ) 
