6 STATEMENT AND EXPOSITION OP 



But r being still = 1.8, the square root of r, or 



rl = 1.34161, 



so that, r being still the leading ratio, the ratio for the interior half-planet Venus, 

 is ri; and this planet furnishes the only existing example of its kind in the plane- 

 tary system. Another will appear in the system of Saturn. 



The relations thus ascertained may be symbolized as follows; the dependence 

 of a following term on that from which it is derived being indicated by a brace 

 connecting the two, and the power of r involved marked outside of the brace: as, 

 for example, we have 



Mars Planet ] ^ Mars \ 



yri ' w. 



Earth (I planet) ] , ^ , . 



rk \ PI. limit }. .{ t, \ Limit (0?) J 



Venus { 1 planet J ^ T^«'*«'^ ^ I r 



Planetary limit Aphelion of Mercury. ' 



(9) This being kept in view, it will be apparent from what precedes, that the 

 rules now established for the derivation of all the distances in the planetary 

 arrangement subsequent to the first, are as follows: — 



[Leading ratio r being =1.8 very nearly] 



Ride 1st. — When the term in question in the series of planetary distances is 

 other than that pertaining to a half-planet, the value of that term may be obtained 

 by dividing the value of the term immediately preceding by the leading ratio. 



Examples. — Thus, as indicated by the symbols, 



Saturn term r -r ■ 

 ; = Mean distance oj Jupiter 



Mars term 



■ = Limit (® 9 ) ; and 



(Q^ g ") 



— ^ — = Aphelion distance of Mercurf. 



[This (incidentally it may be) includes the term for Mercury,^ with the variety, 

 that the term which immediately precedes (and which is to be employed in that 

 computation) is the term pertaining to the half-planet Venus; though Mercury 

 itself is not a half-planet, but even has characteristics approaching to those of a 

 douhle-planet.'\ 



Ride 2d. — The value of any term in the series of exterior half-planets may be 

 obtained by dividing the value of the term immediately preceding that in the 

 planetary arrangements, by r*. 



[The Examples are: The respective mean distances of Uranus and the Eartli, 

 and {he perihelion distance of Mercury. Thus, 



Mars term 



3 = Earth term.'] 



r* -" 



' Incidentally, it may be; for Mercury's mean distance has other relations; as will appear in 

 Section III. 



