Although we most commonly use the notion of a centre or average point with regard to gravity, the same notion is applicable to other cases. Terrestrial gravity is a case of approximately parallel forces, and the centre of gravity is but a special case of the more general Centre of Parallel Forces. Wherever a number of forces of whatever amount act in parallel lines, it is possible to discover a point at which the algebraic sum of the forces may be imagined to act with exactly the same effect. Water in a cistern presses against the side with a pressure varying according to the depth, but always in a direction perpendicular to the side. We may then conceive the whole pressure as exerted on one point, which will be one-third from the bottom of the cistern, and may be called the Centre of Pressure. The Centre of Oscillation of a pendulum, discovered by Huyghens, is that point at which the whole weight of the pendulum may be considered as concentrated, without altering the time of oscillation (p. [315]). When one body strikes another the Centre of Percussion is that point in the striking body at which all its mass might be concentrated without altering the effect of the stroke. In position the Centre of Percussion does not differ from the Centre of Oscillation. Mathematicians have also described the Centre of Gyration, the Centre of Conversion, the Centre of Friction, &c.
We ought carefully to distinguish between those cases in which an invariable centre can be assigned, and those in which it cannot. In perfect strictness, there is no such thing as a true invariable centre of gravity. As a general rule a body is capable of possessing an invariable centre only for perfectly parallel forces, and gravity never does act in absolutely parallel lines. Thus, as usual, we find that our conceptions are only hypothetically correct, and only approximately applicable to real circumstances. There are indeed certain geometrical forms called Centrobaric,[269] such that a body of that shape would attract another exactly as if the mass were concentrated at the centre of gravity, whether the forces act in a parallel manner or not. Newton showed that uniform spheres of matter have this property, and this truth proved of the greatest importance in simplifying his calculations. But it is after all a purely hypothetical truth, because we can nowhere meet with, nor can we construct, a perfectly spherical and homogeneous body. The slightest irregularity or protrusion from the surface will destroy the rigorous correctness of the assumption. The spheroid, on the other hand, has no invariable centre at which its mass may always be regarded as concentrated. The point from which its resultant attraction acts will move about according to the distance and position of the other attracting body, and it will only coincide with the centre as regards an infinitely distant body whose attractive forces may be considered as acting in parallel lines.
Physicists speak familiarly of the poles of a magnet, and the term may be used with convenience. But, if we attach any definite meaning to the word, the poles are not the ends of the magnet, nor any fixed points within, but the variable points from which the resultants of all the forces exerted by the particles in the bar upon exterior magnetic particles may be considered as acting. The poles are, in short, Centres of Magnetic Forces; but as those forces are never really parallel, these centres will vary in position according to the relative place of the object attracted. Only when we regard the magnet as attracting a very distant, or, strictly speaking, infinitely distant particle, do its centres become fixed points, situated in short magnets approximately at one-sixth of the whole length from each end of the bar. We have in the above instances of centres or poles of force sufficient examples of the mode in which the Fictitious Mean or Average is employed in physical science.
The Precise Mean Result.
We now turn to that mode of employing the mean result which is analogous to the method of reversal, but which is brought into practice in a most extensive manner throughout many branches of physical science. We find the simplest possible case in the determination of the latitude of a place by observations of the Pole-star. Tycho Brahe suggested that if the elevation of any circumpolar star were observed at its higher and lower passages across the meridian, half the sum of the elevations would be the latitude of the place, which is equal to the height of the pole. Such a star is as much above the pole at its highest passage, as it is below at its lowest, so that the mean must necessarily give the height of the pole itself free from doubt, except as regards incidental errors. The Pole-star is usually selected for the purpose of such observations because it describes the smallest circle, and is thus on the whole least affected by atmospheric refraction.
Whenever several causes are in action, each of which at one time increases and at another time decreases the joint effect by equal quantities, we may apply this method and disentangle the effects. Thus the solar and lunar tides roll on in almost complete independence of each other. When the moon is new or full the solar tide coincides, or nearly so, with that caused by the moon, and the joint effect is the sum of the separate effects. When the moon is in quadrature, or half full, the two tides are acting in opposition, one raising and the other depressing the water, so that we observe only the difference of the effects. We have in fact—
Spring tide = lunar tide + solar tide;
Neap tide = lunar tide - solar tide.
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.[270] 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.[271] 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.