CERTAIN HARMONIES OF THE SOLAR SYSTEM. 7 



Rule 3cZ. — The value of any term iu the series of interior half-planets may be 

 obtained by dividing the value of the term of the planetary arrangement imme- 

 diately preceding that, by ri. 



\_Examples are: The mean distance of Venus, and that due to the missing interior 

 half-planet, next in the arrangement to the exterior half-planet Uranus. Thus 



Earth term. 



i = Venus term.'] 



With D, or D", or Z)'", as the case may be, for the value of the distance in ques- 

 tion, and D that to which that value is referred, we have 

 For Case under Bide First, 



D' = — ; whence, withal, r = ~-, 

 r 1) 



(a) 



[For Mercury, I)' = -^^l' 

 For Case under Rule S^ond, 



For Case under Ride Third, 







D" = 



D 



lird. 













D"' = 



D 



rh 



e also leai 



n, that 





D' 



, each 



_1 



r 



D" 

 1) 



D" 



1 



r\ ' 

 1 



and 





15 



~ri 







{^) 



(10) These equations express the laws of apportionment of the planetary dis- 

 tances; which are these: — 



Laws of Apportionment of the Planetary Distances. 

 [Value of r = 1.8, very nearly.] 



Law First. For any term subsequent to the first, in the series of terms of plane- 

 tary distances; and other than a half-planetary term: — 



succeeding tertn : prec. term : : 1 : leading ratio r. 



Law Second. For an exterior half-planetary term: — 



ext. half-planet, term : prec. term : : 1 : %power of leading ratio r, i.e. ri. 

 Law Third. For an interior half-planetary term. 



int. half-planet, term. : prec. term : : 1 : square root of leading ratio 9% or r i . 



' {(1) being the term pertaining to the interior half-planet Venus. 



