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and the eye of the spectator, (Fig. 210.) the radius of any arc of which A is the highest point, is equal to the sum of its altitude AOh, and that of the Sun SOH, or hOS. We have therefore only to take with a sextant, or other equivalent instrument, the greatest height of any arc above the horizon, and add that of the Sun, to obtain the radius of the arc.
183. It is sometimes required to determine, from observations on the rainbow, the ratios of refraction, for the different kinds of coloured light, between air and water.
Suppose that we have found the value of θ, or 2φ′−φ for an arc of the primary bow.
| Let tan(2φ′−φ) | = | A, |
| tanφ′ | = | t. |
We saw that in this case
| m·sinφ′ | = | sinφ, |
| and mcosφ′ | = | 2cosφ; |
∴ tanφ′=12tanφ, or tanφ=2tanφ′.
| Then tan(2φ′−φ), or A | = | tan2φ′−tanφ1+tan2φ′·tanφ |
| = | 2t1−t2−2t1+4t21−t2=2t31+3t2; |
∴ 2t3−3At2+A=0.
From this equation we must find t, and from that by a table φ′ and φ: then dividing sinφ by sinφ′, we shall obtain the particular value of m required.
