and the Theory of Saturn’s Rings. 265 
will be expressed by 
aP bQ) cR e 1 
“A? B? "0% e e . e e oy e ( ) 
If N denote the centrifugal force at the extremities of the major 
axis, the intensity at the proposed pomt will be nies , and 
the components in the direction of the three semiaxes will be 
aN bN 
eas, 
To find the components of the disturbing force of the primary 
when the major axis ranges with its centre, we may use methods 
analogous to those pursued in the lunar theory for estimating the 
amount of solar disturbance. Thus, putting # for the radius of 
the circular orbit which the satellite describes, and M for the 
attractive force of the primary at the distance z, the attraction 
which it exerts on the satellite at the point under consideration 
will be 
Oe rc ce 
Mz? 
Pad gt ee ee (3) 
This is equivalent to two forces—one acting in the direction of 
the major axis and expressed by 
M23 
(a®—2aw +02 +b? +42)? 
the other directed to the centre of the satellite and expressed by 
Ma? Va? +02 + c? ; (5) 
(x? —2ax +a? +b? +02)? a eee 
From the first, (4), arises a disturbance operating exclusively 
in the direction of the major axis, and represented by 
pea es gn 8 pn 
(a? —2ax + a? + b?+c?)2 & 
eet 
the squares and higher powers of “» °, and - being rejected as 
too small to affect the result to any appreciable degree. Under 
the same conditions, the radial force (5) resolved with reference 
to the three axes gives the components 
Ma Md Me . 
Ae 3 aaa . e e 4 ) e (7) 
x & 2 
Accordingly, if X represent the sum of the components acting in 
* A demonstration of this theorem may be found in the article on 
“ Attraction ” in the eighth edition of the Encyclopedia Britannica. 
