1875.J 657 [Chase. 



i % 1.6594 cJ^a 1.6444 



J ^^ 10.0113 l2 3 10.0000 



i ©2 .3333 ^1 .3187 



3. Substituting the orbital radius for the radius of linear oscillation, 



nr 2 



we have ; — ~ =r -tt. 



w + 1 3 



§ tp 20.0226 §3 20.0442 



f ©3 -6763 9i .6978 



4. Substituting the radius of incipient aggregation and its correspond- 

 ing radius of linear oscillation, we have 



nr 1.4232 r 



T-T = — 1—\ . • . n^ 2.467 



2 n 4.934 [^/^ 4.978 



The combined influences of Jupiter and Earth over the asteroidal belt, 

 especially as shown in the second and fourth accordances ; the tendency 

 of their mean radial velocities (at 1.4232 r) and the limiting satellite 

 velocities, to equality at Sun's present limiting planetary velocity ; the 

 indications of uniform primitive velocity, furnished by the general pre- 

 dominance of geometrical ratios and the introduction of harmonic values 

 in the minute details ; the a fviori probability of such primitive uni- 

 formity ; the relations of mass and position to orbital times, as well as 

 to atmospheric and nuclear-nebular radii {t, r, and p)-^ all point to origi- 

 nating undulations, propagated, as inferred from the ultimate limit of 

 equality towards which the parabolic cometary and mean radial centrif- 

 ugal velocities both tend, with the velocity of light. 



La Place {Mecanique Celeste, II, viii, 65-69 ; VI, ii, 12-16 ; etc.) investi- 

 gated a number of inequalities depending on the squares and products of 

 the disturbing forces. In his discussions of the Jovian and Saturnian sys- 

 tems he introduced terms containing the 3d and 5th dimensions of the 

 eccentricities and inclinations. The closeness of the agreements here 

 presented may, perhaps, lead to important considerations involving still 

 higher powers. 



If we substitute for the theoretical primitive exponential ratios (1, 1+ 

 2, l-|-2-(-3), the present actual vector radii, (« = tj;^ ; /^ =^ S2 ; ^ = 12 2 ; 

 8 ^ 2/2)) we find an equation for Saturn's mean perihelion : — 



^^XS'^-^X2/«=)2«+'^ (1) 



^2^2 21 ^ ^ ' 



If a, /9, 5, represent the mean aphelion vector-radii, we find an equation 



for Saturn's mean distance : — 



tp^X §^-^XT^^=l2^+'^ (2) 



If we take powers of the masses, instead of powers of the vector-radii 

 equation (2) gives two values for Saturn's mass, according as we use 

 Newcomb's greatest value of Neptune's mass, d^jj-J, deduced from its 

 satellite (3) 



or the least, (1^^00)5 deduced from perturbations of § (4) 



