Dividing by a and inserting from Kepler's first law r a = 1 − − ε ε 2 1 + ε ε cos ⁡ ⁡ θ θ to get cos ⁡ ⁡ E = ε ε + 1 − − ε ε 2 1 + ε ε cos ⁡ ⁡ θ θ cos ⁡ ⁡ θ θ = ε ε ( 1 + ε ε cos ⁡ ⁡ θ θ ) + ( 1 − − ε ε 2 ) cos ⁡ ⁡ θ θ 1 + ε ε cos ⁡ ⁡ θ θ = ε ε + cos ⁡ ⁡ θ θ 1 + ε ε cos ⁡ ⁡ θ θ .