10 STATEMENT AND EXPOSITION OP 



(13) The increment of the leading ratio, or factor r, having been ascertained to 

 be real for the region thus examined, an application of the rule which that implies 

 was tried throughout the planetary system ; and after an enormous number of such 

 tentative processes, the following local values of r were found to give the most 

 consistent results, the values of ?•, it will be seen, increasing withal in arithmetical 

 progression. 



Values of r in the Planetary System. 



Region. Factor r. 



IS eptnne to limit (U) 1.7170 



Limit (U) to Saturn 1.7908 



Saturn to Jupiter ....... . 1.8046 



Jupiter to limit (A) ....... 1.8184 



Limit (A) to Mars 1.8322 



Mars to Zmrt (®9) ...:... 1.8460 



imiV (©9) to the Aphelion of Mercury . . . . 1.8598 



Aphelion of Mercury ]^ , Qirar 



to limit within ) 



0.0138 

 0.0138 

 0.0138 

 0.0138 

 0.0138 

 0.0138 

 0.0138 



The mean of these is 1.8253; differing a little less than -g^gth of itself from either 

 extreme. 



From these we have for the exterior half-planet intervals: — 



Region. Factor rf. 



Neptune to Uranus .......... 1.5369 — 



Mars to Earth 1.5710 — 



Aphelion to Perihelion of Mercury ....... 1.6014 + 



For the interior half-planet intervals, we have: — 



Region. Factor rj. 



Uranus to Si 1.3356 + 



Earth to Venus 1.3612 + 



From the interior half-planet Venus to Mercury 



r= 1.8632 + 



Under these conditions the value of the half-planet limit S«, i.e. interior to 

 Uranus, may now be determined; and it will be found to be 14.64275.^ 



(14) The arrangement of the planetary system in accordance with all that has 

 now been determined, is similar to that of the Second Approximate Arrangement 

 heretofore exhibited, (10); the value of the interior half-planet limit Si and the 

 standard value^ of the asteroid limit (A) being both inserted; and besides the column 

 of differences of Law from Fact in terms of the Earth's mean distance as 1, we have 



would then approach more nearly to the standard value of limit (A). In this aspect of the matter, 

 the difference of limit (A) from the mean in question would seem to be on the right side. 



If, however, we take the mean betvfeen the two extremes of the known distances, that of Flora 

 2.20336, and that of Sylvia 3.49411 (as Prof. Hirkwood has done — Proceed, of Boyal Ast. Soc, 

 vol. xxix. p. 99), we shall have the value 2.84873 ; which is almo.st exactly the same with the 

 value of (A) here brought out. 



^ What ought to be the mass of the missing half-planet cannot be ascertained without the intro- 

 duction of theoretical considerations ; of which more hereafter. 



° As exhibited in Article (12). 



