1875.] U55 [Chase. 



and ^ Mercury. If we take seven geometrical means between ^ Mer- 

 cury's, and Neptune's, mean radius-vector, we fiad the following accord- 

 ances : 



3 ©3 

 * * 



i §, 



The geometric ratio of the theoretical column is 1.879, or almost pre- 

 cisely the sum of the co-efficients of the Urano-N"eptunian belt (| + f ). 

 It will be observed that the theoretical co-efficients (^ 9 > I © • • • I S ) 

 are the same as appear in the inter-planetary abscissas of my CentauruF- 

 Heliacal parabola.* The collisions of particles, in their approach to the 

 focus of a paraboloid, would naturally convert pai-abolic into elliptical 

 orbits ; and particles falling towards a cosmic focus from a distance ni' 

 would acquire the dissociative velocity (relatively to the Sun) y'2~gr, at 



nr 

 — — ^ from the focus. By giving to n, successive values in arithmetical 



progression, we form the arithmetico-harmonic series, j f f | -| f h 

 which constitute the j)eculiar sequence of co-efficients, both in the fore- 

 going geometric series and in the abscissas of the primitive parabola. 



The bases of the principal planetary harmonies that have been hitherto 

 published, are : — Peirce (phyllotactic), the time of orbital revolution, t ; 

 Bode, and Alexander, the orbital radius vector, or the radius of possible 

 solar-nebular atmosphere, r ; my own (harmonic), the nucleal radius, p. 

 Their common relations may be thus shown : — 



1 J 2 



Peirce, t oc t oc r '^ cc p 



a 1 4 



Bode, Alexander, r cc i'-^ oc r oc p'^ 



i 3 1 



Chase, p oc t- oc r '^ oc p 



The Saturnian relations of inertia seem to have established the Bode 

 series. For if we take as our unit, /?g=: 20.58 solar radii, (p^ being 

 the nucleus of a nebulous sun which would rotate synchronously with Sat- 

 urn's orbital revolution), we obtain the following values : — 



