Note that when β = 1, hence, cosσ = ±1= cosσ'_i, when positive σ = 0º (0) = σ'_i and when negative σ = 180º (π) = σ'_i. Also when β is zero, cosσ = 0 = cosσ'_i and thus when δp and δip' are positive σ = 90º (½π) = σ'_i (and cscσ = 1 = cscσ'_i), while when δp and δip' are negative σ = 270º (3π/2) = σ'_i (and cscσ = -1= cscσ'_i). (The range of the cosecants in both equation [7] and [10] is ±∞)