418 Mr. A. Cay ley on the Cubic Centres of a Line 



of e is between 87 and 88. From the corresponding value of 



R 3 



-j-, it appears that x at this critical period is equal to 2*489 R. 



This expresses approximately the least distance from the primary 

 at which a satellite could be preserved. To give greater gene- 

 rality to the formula, designate by D and d the densities of both 

 bodies, r' being, in accordance with the previous notation, equal 

 to the actual radius of primary, and R the radius of the sphere 

 which it might fill if it became as dense as the satellite. It will 



3 /5" 



be easily seen that R=? j a/ -r, and accordingly x' is equal to 



2-489 r^! (31). 



It appears also to be independent of the actual size of the 

 satellite. If indeed we retained in our investigation the square 

 and the higher powers of r, the result would exhibit a slight 

 difference in favour of the stability of smaller attendants ; but 

 this is more than counterbalanced by the superior density of 

 larger bodies composed of the same materials, but capable of 

 compressing them by a greater attractive force. 



It appears, therefore, that the laws of equilibrium prevent the 

 existence of satellites over a large space around each primary 

 planet ; and it might also be shown that, beyond this region, the 

 existence of planetary rings is equally impossible. The extent 

 of this region will be different for satellites unequally dense, and 

 it may be easily calculated in each case by our last formula. 

 Although our data is somewhat defective in the case which pre- 

 sents itself in the Saturnian system, yet when we calculate the 

 density which a satellite should possess to maintain its planetary 

 form in different parts of the zone occupied by his rings, it seems 

 impossible to resist the conclusion that the condition of this an- 

 nular appendage is the necessary consequence of its proximity 

 to Saturn. 



Cincinnati, October 29, 1859. 



LVI. On the Cubic Centres of a Line with respect to Three Lines 

 and a Line. By A. Caylby, Esq* 



CONSIDER a line L in relation to the three lines X, Y, Z 

 and the line I : through the point of intersection of the 

 lines X, L, draw any line meeting the lines I, Y, Z, and let the 

 harmonic of the intersection with I, in relation to the intersec- 



* Communicated by the Author. 



