F₁ = v₁/(gy₁)^1/2

F₂ = v₂/(gy₂)^1/2

From continuity:

v₁y₁ = v₂y₂

v₁²y₁² = v₂²y₂²

F₁²y₁³ = F₂²y₂³

F₂² = F₁²/ (y₂/y₁)³

Eq. 3-21 (Chow):

y₂/y₁ = (1/2)[(1 + 8F₁²)^1/2 - 1]

N² = 1 + 8F₁²

y₂/y₁ = (1/2) [N - 1]

2 (y₂/y₁) = N - 1

(y₂/y₁)³ = (1/8) [N - 1]³

2 (y₂/y₁)³ = (1/4) [N - 1]³

N = (1 + 8F₁²)^1/2

N³ = (1 + 8F₁²)^3/2

N² - 1 = 8F₁²

F₁² = (N² - 1)/8

4F₁² = (N² - 1)/2

E₂/E₁ = [y₂ + v₂²/(2g)]/[y₁ + v₁²/(2g)]

E₂/E₁ = [y₂ (1 + F₂²/2) ]/[y₁ (1 + F₁²/2)]

E₂/E₁ = 2 (y₂/y₁) {1 + F₁²/[2 (y₂/y₁)³]}/(2 + F₁²)

E₂/E₁ = (N - 1){1 + (N² - 1)/[2 (N - 1)³]}/(2 + F₁²)

E₂/E₁ = (N² - 1)(N - 1) {1 + (N² - 1) / [2 (N - 1)³]}/[8 F₁² (2 + F₁²) ]

E₂/E₁ = [(N² - 1)(N - 1) + (1/2)(N + 1)²]/[8 F₁² (2 + F₁²)]

E₂/E₁ = {(N³ - N² - N - 1) + [(N²/2) + N + (1/2)]}/[8 F₁² (2 + F₁²)]

E₂/E₁ = {(N³ - [(N² - 1)/2] + 1}/[8 F₁² (2 + F₁²)]

E₂/E₁ = [(8F₁² + 1)^3/2 - 4 F₁² + 1]/[8F₁²(2 + F₁²)]