CERTAIN HARMONIES OF THE SOLAR SYSTEM. 77 



[With a whole-planet arrangement of the asteroid-mass, the resulting time of 

 rotation of Mars would be 19/i.968; the half-planet arrangement of (60), thus 

 appearing again as preferable.] 



So that, in the case of the asteroids, although the component material has been 

 dispersed; yet, as a half-planet portion has not passed over and been absorbed by 

 an interior planet, the determining conditions of the next interior planet's rota- 

 tion have, it would seem, not been entirely disturbed. 



Of '■'■ Bode's Law" and the reasons for its success in tJie approximate determination 

 of the respective distances of Uranus and several other planets^ and also for its 

 failure to determine the distance either of Saturn or that of Neptune. 



(109) The most simple statement of the (so-called) Law of Bode (or of Titius) 

 is that of Sir J. Herschel; viz.: .... "The interval between the orbits of the 

 !Earth and Mercury is nearly twice that between those of Venus and Mercury; that 

 between the orbits of Mars and Mercury nearly twice that between the Earth and 

 Mercury; and so on."^ 



Now, (13), the mean value of our whole-planet ratio is (stated here approxi- 

 mately) 1.8. But, if we subtract Mercury's distance from each of two successive 

 terms in the whole-planet series, to obtain the intervals between orbits here in 

 question, the ratio of the remaining intervals will exceed the ratio r of 1.8 + , since 

 the smaller of the two distances compared will be more than proportionally dimin- 

 ished by such a subtraction ; and the value of greater divided by the less (i. e. here 

 of the ratio) will be increased. Thus : — 



Asteroid limit (A) 

 . Mars^ distance 



But 



(A) — Mercunfs distance 



= 1.8+ 



2+; 



Mars^ — Mcrcuri/s distance 



the ratio being a very little greater than that which " Bode's Law" requires. 



The same ratio is, even, very well justified in the instance of the Earth compared 

 with Venus, and Mars with the Earth ; though [as exhibited in Table (B) in (14)], 

 while the ratio of the distance of Venus to that of Mercury is (incidentally) the 

 whole-planet ratio r, that of the Earth's distance to that of Venus is only o-i, and 

 even the ratio of Mars' distance to that of the Earth is only ri. But the increase 

 of the measuring unit in the comparison, as we proceed, and the subtraction of 

 Mercury's distance in every instance (one being more effective in the one case, and 

 the other, in the other) together make the one interval near to the double of the 

 other. 



The ratio, as has been already stated, nearly accurate for the Astexoid-intervcd in 

 the middle of the whole-planet series. But, when we pass beyond that to the Jupiter 

 and Saturn terms, successively, the subtraction of only Mercury's distance, though 

 just about sufficient for the justification of the Jupiter interval, gives a result too 

 small in the instance of that of Saturn. 



' Outlines of Astronomy (11th Edition), (505) 



