CERTAIN HARMONIES QF THE SOLAR SYSTEM. 9 



in which we possess three out of the four requisite terms ;^ the fourth (the asteroid 

 term or limit (A)) to be accurately determined by the process here proposed, and 

 its value thus obtained to be made the criterion for the comparison of its value as 

 ascertained in the more extended series. In the several instances of the three 

 planets here in question, there are withal no half-planet relations, and the fourth 

 term being a limit in the regular series in which r enters, the half-planet relation 

 does not pertain to it ; so that the character of the leading factor r, as to variability 

 or otherwise, is here to be sought for. 



(12) Now the existing mean distances from the sun in this region, together Avith 

 the asteroid limit (A), may be arranged as follows, viz. : — 



Dist. from Suu.^ Log. of Ratios. Differeuce. 



Saturn .... 9.53885 + „ 9G32591 



Jupiter .... 5.20280 o'2(;55331- + 0.0022 Y4 



Limit {X) .... (2.8229G-) 02678071- 0.002274 



Mars 1.52369 + 



The log. differences being equal, the ratios themselves increase in geometrical 

 progression. 



But if the arrangement be made with the ratios increasing in ai-ithmetical pjro- 

 gression, we shall have — 



Dist. from Sun.^ Ratios. Difference. 



Saturn , . . . 9.53885 + 



Jupiter .... 5.20280 1.83341 _^ 0.00964 



Limit (A) .... (2.82293—) 1-84305 0.00964 



Mars .... 1.52369+ 1-85269 



Now we do not know enough of the nature of the case to decide which of these 

 conditions ought to prevail, though the analogy of logarithms etc. would lead us 

 to suppose that the ratios themselves should increase in arithmetical progression. 

 But, happily, such a decision is of no moment practically ; since the differences in 

 question are so small, that the value of the limit (A) in the one case differs from 

 that in the other only in the fifth decimal place. 



So the value of the limit (A) = 2.82293-, which is that due to the increase of 

 the ratio in arithmetical progression, will be accepted, and the same will be adopted ; 

 and then, as heretofore intimated, this value will be made the criterion for the com- 

 parison of the value as ascertained in the more extended series. This standard 

 value, being withal a direct derivation from fact, in its own special region, will here- 

 after be inserted as a limit in the column of Fact, the figures being inclosed in a 

 parenthesis.* 



' In the order of discovery, it was in thii^ region that the appro.Yimation of the series of distances 

 to a geometrical progression, with the ratio = 1.8 nearly, was first discerned. 



» See Table (A), in (3). 



^ This value, 2.82293, is greater than the mean of the distances from the sun of 122 known asteroids, 



which is only 2.70282. But then about /^ of that number are distances below the mean; leaving 



but -,\ above the same. So that it seems not unreasonable to suppose that were many more included, 



which mostly are now unknown — partly, it may be, because of their greater distance — the mean 



2 November, 1S74. 



