in which l = the distance in English miles L′d at which the light would strike the ocean’s surface. We then reduce this value of H′ by the correction for mean refraction, which permits the light to be seen at a greater distance, and which =
2 l² 21, (2.)
So as to get, H = 2 l²3 - 2 l²21 = 4 l²7 (3.)
an expression which at once gives the height of the tower required, if the eye of the mariner were just on the surface of the water at d, where the tangent between his eye at S and the light at L would touch the sea. We must, therefore, in the first instance, find the distance d S = l′, which is the radius of the visible horizon due to the height SS′ = h of his eye above the water, and is, of course, at once obtained conversely by the expression:—
l′ = √7 h 2 (4.)
Deducting this distance from SL, the whole effective range of the light, we have L d = l, and operating with this value in the former equation,
H = 4 l²7
we find the height of the tower which answers the conditions of the case.[81] From the above data the following Table has been computed.
[81] In the above expressions l and l′ are given in English miles, which in Scotland may be considered as bearing to nautical miles the ratio of 5280 to 6088. In order to convert a distance given in nautical miles to English miles, all that is needful is to add the log of the number of nautical miles to log 5280, and subtract log 6088.
| H | λ | λ′ | H | λ | λ′ | H | λ | λ′ |
|---|---|---|---|---|---|---|---|---|
| Heights in Feet. | Lengths in English Miles. | Lengths in Nautical Miles. | Heights in Feet. | Lengths in English Miles. | Lengths in Nautical Miles. | Heights in Feet. | Lengths in English Miles. | Lengths in Nautical Miles. |
| 5 | 2·958 | 2·565 | 70 | 11·067 | 9·598 | 250 | 20·916 | 18·14 |
| 10 | 4·184 | 3·628 | 75 | 11·456 | 9·935 | 300 | 22·912 | 19·87 |
| 15 | 5·123 | 4·443 | 80 | 11·832 | 10·26 | 350 | 24·748 | 21·46 |
| 20 | 5·916 | 5·130 | 85 | 12·196 | 10·57 | 400 | 26·457 | 22·94 |
| 25 | 6·614 | 5·736 | 90 | 12·549 | 10·88 | 450 | 28·062 | 24·33 |
| 30 | 7·245 | 6·283 | 95 | 12·893 | 11·18 | 500 | 29·580 | 25·65 |
| 35 | 7·826 | 6·787 | 100 | 13·228 | 11·47 | 550 | 31·024 | 26·90 |
| 40 | 8·366 | 7·255 | 110 | 13·874 | 12·03 | 600 | 32·403 | 28·10 |
| 45 | 8·874 | 7·696 | 120 | 14·490 | 12·56 | 650 | 33·726 | 29·25 |
| 50 | 9·354 | 8·112 | 130 | 15·083 | 13·08 | 700 | 35·000 | 30·28 |
| 55 | 9·811 | 8·509 | 140 | 15·652 | 13·57 | 800 | 37·416 | 32·45 |
| 60 | 10·246 | 8·886 | 150 | 17·201 | 14·91 | 900 | 39·836 | 34·54 |
| 65 | 10·665 | 9·249 | 200 | 18·708 | 16·22 | 1000 | 41·833 | 36·28 |