CERTAIN HARMONIES OF THE SOLAR SYSTEM. 37 



Then for the mass of the interior half-planet ^i, we need first to redistribute the 

 material of Mercury, so as to place its whole-jylanet portion at the aphelion, and its 

 half-planet portion at the perihelion ; to come back to the forming state, etc., 

 described find exhibited in symbol in (50). 



Putting then the whole mass of Mercury ^ to 1 ; if that be so distributed to 

 the aphelion and perihelion positions, that the centre of gi/ratioa of the distributed 

 portions shall be found at Mercury's mean distance,^ we shall have — 



0.5617245 of Mercury's mnss, for the aphelion, and 

 0.4382755 " " " " " perihelion. 



The values thus far requisite having been ascertained, the case is but a repeti- 

 tion of that of the mass of Si in (41) ; and by substituting the values now before 

 us, and reducing, we shall find the value of the mass of the interior half-planet — 

 m of 5i, interior to Mercury, =0.594059 of the mass of Mercury. 



(52) Now M. Le Verrier, in the Co^nptes Fendus, tome XLIX. p. 382, (Sept. 

 1859), speaking of a cause adequate to produce an ascertained secular motion of 38" 

 in the perihelion of Mercury, admits the supposition of a hypothetical planet, 

 situated between Mercury and the Sun, and says that, as the hypothetical planet 

 ought to impress on the perihelion of Mercury a secular motion of 38 seconds, the 

 resulting relation between its (the planet's) mass and its distance from the sun will be 

 such that, in measure, as we suppose the distance less, the mass will be increased, 

 and the converse : and he adds, that, " For a distance a little less than the half 

 of the mean distance of Mercury from the Sun, the mass sought would be equal 

 to that of Mercury." 



The mass which, on our own ^j?rt?i, in the ftjlloioing out of our own hypothesis, (51), 

 we have found for the hypothetical planet is 0.594059 of the mass of Mercury ; 

 and when, in conjunction with Mercury, as seen from the sun, the distance between 

 the two planets [see (51) and Table (A), in (3)], would be 



0.38710 — 0.20836 = 0.17874 ; 



and " a mass equal to that of Mercury," similarly situated, would have the same 

 attractive force with that due to our hypothetical planet, at a distance, for that 

 mass, inside of Mercury = to 0.23190, i. e., a distance from the sun =0.15520; 

 which is indeed, assuredly, somewhat " less than the half of the mean distance of 

 Mercury from the Sun," which | distance, accurately, ^0.19355. 



^ For this purpose, m-\-vi', the sum of the two masses, being put^ to 1 ; m'==l — m. 



Also — since the ratios of the distances are known, or may be readily ascertained — if (C) be the 

 distance of the centre of gyration, and the distance of the outer body= q (C), and that of the inner 

 =p (C); then, substituting in Eq. (C) in (17), and reducing, we shall have, for the fraction of the 

 whole mass pertaining to the inner body, 



ni^-^ , : 



T — P' 



which will, also by substitution and subtraction, give us m', since it = 1 — m. 



