a value between 0 and 1. The potential function can then be written as , (28) ))/(exp()/exp(2eegrrrc-=Fwhich can be expanded as L+-+++=F332222)4/1(1)/exp(rrwrrwrrceee, (29) where )2/1(+=gw. (30) With this form for the potential, and using 220~ercme-= from (23), (27) can be expanded as L4443444214342143421434213214342143421321+-+-+++--+=765432103322023042022220202230420220)4/1(~~8~2~~2~8~2~EeEEeEeEeEEErrwcmrcmreprmrpwrcmrwrmrpcmprempcmE (31) Each of the braced terms has a specific physical meaning, and each one can be quantized according to the unperturbed Bohr theory (see section 8.3.3 of Ref. 3). We then find the following expressions: 200~cmE= : rest mass energy L,2,1,/21=-=nnREe : Bohr energy nknknREeL,2,1,431222=......--=a : relativistic correction of the Bohr- Sommerfeld model ......-=nknREe12323a : orbital magnetic energy kncmekncmwknRwEe32408320432412~1~12h===aa : quantum mechanical spin energy 20240425~~221................=mmrcmevE : radiative reaction The constant 2/~202cmRea= is the Rydberg constant,p2/h=h, and is the fine-structure constant. The value , which corresponds to the quantum values of electron spin, arises from truncating the series after E6, which implies setting E7 and higher order energies equal to zero. In physical terms this implies a limit to the resolution of our observational apparatus. ceh/2=a12w= The sum of the first three terms gives the fine-structure spectrum of the hydrogen atom: ............-+-.++nknncmEEE431121~/)(22220210aa . (32) The sum of the remaining terms E3 and E4 represents the total spin-orbit energy. This energy is related to higher order effects such as the Lamb shift. Details about these effects can also be found in Ref. 3. The calculated value of the Lamb shift differs by an order of 10 from the observed value. However, since the Lamb shift is a third order perturbation of the hydrogen spectrum, the discrepancy could possibly be resolved by a second order correction to the effective nuclear radius. For k=1 the expression for E4 is equal to the Darwin term of the Dirac theory.6 9. The Large Number Coincidences Dirac postulated that the large dimensionless ratios (~1040) of certain universal constants underlie a fundamental relationship between them. A theoretical explanation for these ratios has not yet been found, but it became known as Diracs Large Number Hypothesis (LNH).7 Some of these relations are derivable from (24). Taking R as the Schwarzschild radius of the proton, Rp = 2GMp/c2, we rewrite (24) as ,~2202cGMcmep=- or .1~02=-mGMep (33) Defining the relationship between the gravitational mass Mp of a proton and its electromagnetic rest mass pm0~, as pDpmNM0~= , (34) where ND is a dimensionless number, we can write (33) as DpNmmGe=-002~~ . (35) Taking the absolute value of the above equation gives the basic relationship of the LNH. Further LNH derivations are derived in Ref. 3. 10. A unified force model One can derive the forces acting on a particle by applying the Euler-Lagrange equations to the Lagrangian of the system. From the Lagrangian of (19) we find a Lorentz-type force equation (see 7.4 of Ref. 3): ][~0HvEkmprrr&r+= , (36) where for Gravitation: 1-=k, Electromagnetism: 1+=k. The same equation gives either planetary or atomic motion, where the vectors Er and Hrare respectively given by 222rcrrrGMrEe==r , (37) 332)()(rrvrrcrvGMHerrrrr== . (38) 11. Interim summary We have shown in a descriptive way that the Newtonian-type Lagrangian (1) gives both the gravitational and electromagnetic