We have only then to add together the heights of the maximum spring tide and the minimum neap tide, and half the sum is the true height of the lunar tide. Half the difference of the spring and neap tides on the other hand gives the solar tide.
Effects of very small amount may be detected with great approach to certainty among much greater fluctuations, provided that we have a series of observations sufficiently numerous and long continued to enable us to balance all the larger effects against each other. For this purpose the observations should be continued over at least one complete cycle, in which the effects run through all their variations, and return exactly to the same relative positions as at the commencement. If casual or irregular disturbing causes exist, we should probably require many such cycles of results to render their effect inappreciable. We obtain the desired result by taking the mean of all the observations in which a cause acts positively, and the mean of all in which it acts negatively. Half the difference of these means will give the effect of the cause in question, provided that no other effect happens to vary in the same period or nearly so.
Since the moon causes a movement of the ocean, it is evident that its attraction must have some effect upon the atmosphere. The laws of atmospheric tides were investigated by Laplace, but as it would be impracticable by theory to calculate their amounts we can only determine them by observation, as Laplace predicted that they would one day be determined.10 But the oscillations of the barometer thus caused are far smaller than the oscillations due to several other causes. Storms, hurricanes, or changes of weather produce movements of the barometer sometimes as much as a thousand times as great as the tides in question. There are also regular daily, yearly, or other fluctuations, all greater than the desired quantity. To detect and measure the atmospheric tide it was desirable that observations should be made in a place as free as possible from irregular disturbances. On this account several long series of observations were made at St. Helena, where the barometer is far more regular in its movements than in a continental climate. The effect of the moon’s attraction was then detected by taking the mean of all the readings when the moon was on the meridian and the similar mean when she was on the horizon. The difference of these means was found to be only ·00365, yet it was possible to discover even the variation of this tide according as the moon was nearer to or further from the earth, though this difference was only ·00056 inch.11 It is quite evident that such minute effects could never be discovered in a purely empirical manner. Having no information but the series of observations before us, we could have no clue as to the mode of grouping them which would give so small a difference. In applying this method of means in an extensive manner we must generally then have à priori knowledge as to the periods at which a cause will act in one direction or the other.
We are sometimes able to eliminate fluctuations and take a mean result by purely mechanical arrangements. The daily variations of temperature, for instance, become imperceptible one or two feet below the surface of the earth, so that a thermometer placed with its bulb at that depth gives very nearly the true daily mean temperature. At a depth of twenty feet even the yearly fluctuations are nearly effaced, and the thermometer stands a little above the true mean temperature of the locality. In registering the rise and fall of the tide by a tide-gauge, it is desirable to avoid the oscillations arising from surface waves, which is very readily accomplished by placing the float in a cistern communicating by a small hole with the sea. Only a general rise or fall of the level is then perceptible, just as in the marine barometer the narrow tube prevents any casual fluctuations and allows only a continued change of pressure to manifest itself.