The existence of global solution of the n-body problem

Based on McGehee's transformation f = I^-1q; g = I ^{1 2p};, dt dt′ = I ^{3 2}, I introduce the transformtion X = (g^TM^-1g) ^{1 2}, h ≥ 0X = (g^TM^-1g) - 2Ih) ^{1 2}, h≤0, u = IX^-2; F = X²f; G = X^-1g; dt′ dt = X^-3. I prove that these variables may be continued to every point of the new time axis t for any initial value, and the whole axis corresponds to the "time interval of existence of the global solution". Also, F, G, u are O(e^Bτ). I then obtain a region H on the complex plane τ, |Im(τ)| < A exp (B Re τ²), over which F and G are analytic. Here, A, B, C are constants related only to the masses and the initial value. Lastly, a conformal mapping is established which maps a subregion of H, H, onto the unit circle of the new complex variable, thus obtaining a global solution of the n-body problem. The convergence of my power series is admittedly unsatisfactory and so the present result is of limited value for practical calculation.