THE SATUKNIAN SYSTEM. 



BY BENJAMIN PEIRCE. 



The ring is the characteristic feature of the Saturnian system, and its theory is 

 the chief object of the present investigation. Observation has shown the ring of 

 Saturn not to be one, but to be subdivided into two or, possibly, more concentric rings. 

 Tbe mutual action of these rings cannot be neglected in the complete theory, nor the 

 reciprocal action of the rings and the satellites. But the analysis properly commences 

 with the consideration of a single elementary ring. The actual ring is thin, which 

 facilitates the computations; but it is proposed to extend this memoir, which must be 

 regarded as merely introductory, to the research into the attractions of any ring, how- 

 ever thick. Such a ring is, geometrically, a hollow cylinder, and the resulting formulae 

 will be found to be curious and novel aggregates of discontinuous terms, in which the 

 discontinuities unexpectedly balance each other and leave an uninterrupted continuity 

 in the numbers deduced from the formulae. 



T. Tbe ring is assumed to be of uniform density and thickness, and to fill the 

 space contained between two co-axial and concentric circular cylinders. The following 

 notation is adopted : — 



2 b = the length of the cylinder or the thickness of tbe ring, 

 « 2 = the radius of the base of the outer cylinder, 

 «! = the radius of the base of the inner cylinder, 



r = the distance of tbe attracted point from the axis of the cylinder, 

 Q = the distance of a particle of the ring from the axis, 

 ([ = the angle from /• to o. 



By the equatorial plane, or simply the plane of the ring, is to be understood the 



plane which is drawn perpendicular to the axis and midway between the bases. 

 34 ID (263) 



