120 Mr. J. Utting on aPlandaiij Analogij 



of the planets, and their respective systems of satellites, ex- 

 tending to the whole planetary system, resulting from the 

 periodic times and mean distances of the planets, with the 

 periodic times and mean distances of their satellites, com- 

 pounded with the attractive power of the sun, as compared 

 with that of the )irimary planets, around which each respec- 

 tive system of satellites circulates ; viz. 



Let V, V, V", &c. represent the velocities of the planets in 

 their orbits; and a/D, a/D', a/ D", &c. the square roots of tlieir 

 mean distances from tlie sun. Let also v, v\ v", &c. repre- 

 sent the velocities of their respective satellites; and the 

 V^j Vd', A/d", &c. the square roots of their mean distances 

 from their primaries. Let the square root of the sun's attrac- 

 tive power, that of each i>lanet being unity, be denoted by 

 x^m, A/m\ Vm", &c. respectively. 



Whence we have V x VD = V x ^/D' = V" X -/D" &c. a 

 constant quantity for the primarj^ planets. And v x s/ d = 



V X \/d' = v" X y^d', &c. a constant quantity for each respective 

 system of satellites. Also, v x ^d x ^m.-=v' x ^d' x y/in.= 

 rt' X ,s/d" X a/zm", &c. a constant quantity equal to that in the 



first analoffv. Whence -^ = 0. = — ' = 



*'•' V X ^d X 'Jm V X a/cI X V" 



o.= -5:::^5i_=o.&c. 



v" X Vd" X -\/m" 

 The following general analogy also obtains, viz. As 



V : V : : ^/D' : ^/D; also as v:v':: s/d' : ^d &c. where the 

 product of the two extreme terms will always be equal to the 

 product of the two mean ones, whatever may be the planets 

 or satellites fixed on. 



The following table exhibits the result of my calculations 

 in elucidation of this analogy. 



Tabular View of the Analogy rv/iich obtains in the Planetary 

 System. 



iSrxtd- 



