[66] The truth of the first of these equations (sin ξ = m . sin γ) which merely expresses the ratio of the sines of the angles of incidence and refraction is obvious; but owing to the great number of small angles about C, a little consideration may be required to enable one to perceive the truth of the second. I therefore subjoin the steps by which I reached it. It is obvious (see [fig. 72]), that as ACH and BCF are equal, the line SC bisecting HCF must bisect ACB. But the production of AC clearly gives SCD opposite and equal to ACW and SCD is by construction = (α - ξ + γ) = (α + γ - ξ), and, therefore, ACB, which is twice ACW or SCD = (2 α + 2 γ - 2 ξ). Now, by construction OC is a normal to the refracting surface CB and its production C g gives AC g = γ. But γ = ACB - g CB = (2 α + 2 γ - 2 ξ) - g CB = (2 α + 2 γ - 2 ξ) - 90°, hence
| γ | = | {2 α + 2 γ - 2 ξ} - 90°, |
| and γ - 2 γ | = | -γ = -2 ξ + (2 α - 90°) by transposition, and finally changing signs, we have as above: |
| γ | = | 2 ξ - (2 α - 90°) |
| = | 2 ξ - θ. |
Eliminating γ between these two equations we obtain:
sin ξ = m . sin (2 ξ - θ)
an expression, which, after various transformations of circular functions, assumes the form
sin⁴ ξ - 1m sin θ . sin³ ξ + (1 4 m² - 1) . sin² ξ + 12 m sin θ . sin ξ + ¹⁄₄ sin² θ = 0[67]
[67] This expression is equivalent to that of M. Fresnel, but owing to a simplification in the fractional coefficients, it is not literally the same. I was led to it by the following steps, starting from the original equation sin ξ = m sin (2 ξ - θ)
| sin ξ | = | m sin (2 ξ - θ) |
| = | m{sin 2 ξ . cos θ - cos 2 ξ sin θ} | |
| = | m cos θ . sin 2 ξ - m sin θ . cos 2 ξ | |
| = | m cos θ . 2 sin ξ . cos ξ - m sin θ . {1 - 2 sin² ξ} | |
| = | 2 m cos θ . sin ξ . cos ξ - m sin θ + 2 m sin θ . sin² ξ. |
Therefore, m sin θ + sin ξ - 2 m sin θ sin² ξ = 2 m cos θ . sin ξ . cos ξ.
Then: