Differentiating u twice and making use of the Pythagorean identity gives D d 2 u d θ θ 2 = sec ⁡ ⁡ θ θ tan 2 ⁡ ⁡ θ θ + sec 3 ⁡ ⁡ θ θ = sec ⁡ ⁡ θ θ ( sec 2 ⁡ ⁡ θ θ − − 1 ) + sec 3 ⁡ ⁡ θ θ = 2 D 3 u 3 − − D u .