On the Stability of Satellites in small Orbits. 263 
the several coefficients bemg, it will be observed, cyclical func- 
tions to the cycle 4, c, e, d. 
Putting for a its value —(x—6), and for 4, c, d, e their values, 
the quintic equation in z is 
| (x—0)° 7 
+ (z—9)°. —5(AC+BD)28y6 | 
+(z—@)?. —5(A*Byd+ B*Cde2+C*D28 + D*ASy)a8ys 
74 5(A®D 0776 + BeAyé*2 + C#Bda28 + D5C28*y)a8yd = 
+5(A?C?+ B*D?—ABCD)2*6*°e . ae. 
J 
(A588? + Beyé8x2 + 058238? + Dea S52) 288 
—5(ASBCyS + B°CDaz + C3D Az + DSABBy)a262y°S2 
4.5(AB°C228 + BC*D2@2 + CD2A2y8-+ DE2A%S2) e262 
where, as before, 
A=K-+La+My+Nay, 
B=K+L8+Mé6+N86, 
C=K+ly+Ma+Nye, 
D=K+L6+MS8+N68; 
and the coefficients of the quintic equation are, as they should 
be, cyclical functions to the cycle z8y6. 
2 Stone Buildings, W.C.., 
February 10, 186]. 
XLII. On the Stability of Satellites in small Orbits, and the 
Theory of Saturn’s Rings. By Dantet VauGHAn *. 
is dee mysterious revolutions of planets and comets were not 
rendered intelligible to astronomers until mathematical 
investigations revealed the peculiar curves which moving bodies 
must describe when left to the exclusive control of solar gravity. 
The process of deductive inquiry, which proved so beneficial in this 
and other departments of celestial mechanics, may be successfully 
applied to another problem which the results of telescopic obser- 
vation have forced on the attention of mathematicians. The 
physical constitution of Saturn’s rings, the circumstances on 
which their stability depends, and the causes which prevent their 
conversion into satellites, have already been made the subject of 
many able essays; but, though regarding these productions as 
valuable contributions to science, I think it advisable to select a 
* Communicated by the Author. 
