points of three excursions of the beam from the zero of the scale, then (a + b) will be about as much erroneous in one direction as (b+c) in the other, so that the mean of these two means, or (a + 2 b + c), will be exceedingly near to the point of rest.1 A still closer approximation may be made by taking four readings and reducing them by the formula (a + 2 b + 2 c + d).

this

The accuracy of Baily's experiments, directed to determine the density of the earth, entirely depended upon mode of observing oscillations. The balls whose gravitation was measured were so delicately suspended by a torsion balance that they never came to rest. The extreme points of the oscillations were observed both when the heavy leaden attracting ball was on one side and on the other. The difference of the mean points when the leaden ball was on the right hand and that when it was on the left hand gave double the amount of the deflection.

A beautiful instance of avoiding the use of a zero point is found in Mr. E. J. Stone's observations on the radiant heat of the fixed stars. The difficulty of these observations arose from the comparatively great amounts of heat which were sent into the telescope from the atmosphere, and which were sufficient to disguise almost entirely the feeble heat rays of a star. But Mr. Stone fixed at the focus of his telescope a double thermo-electric pile of which the two parts were reversed in order. Now any disturbance of temperature which acted uniformly upon both piles produced no effect upon the galvanometer needle, and when the rays of the star were made to fall alternately upon one pile and the other, the total amount of the deflection represented double the heating power of the star. Thus Mr. Stone was able to detect with much certainty a heating effect of the star Arcturus, which even when concentrated by the telescope amounted only to o°02 Fahr., and which represents a heating effect of the direct ray of only about 000000137 Fahr., equivalent to the heat which would be received from a three-inch cubic vessel full of boiling water at the distance of 400 yards. It is probable that

2

1 Gauss, Taylor's Scientific Memoirs, vol. ii. p. 43, &c.

Proceedings of the Royal Society, vol. xviii. p. 159 (Jan. 13, 1870). Philosophical Magazine (4th Series), vol. xxxix. p. 376.

Mr. Stone's arrangement of the pile might be usefully employed in other delicate thermometric experiments subject to considerable disturbing influences.

Determination of Maximum Points.

We employ the method of means in a certain number of observations directed to determine the moment at which a phenomenon reaches its highest point in quantity. In noting the place of a fixed star at a given time there is no difficulty in ascertaining the point to be observed, for a star in a good telescope presents an exceedingly small disc. In observing a nebulous body which from a bright centre fades gradually away on all sides, it will not be possible to select with certainty the middle point. In many such cases the best method is not to select arbitrarily the supposed middle point, but points of equal brightness on either side, and then take the mean of the observations of these two points for the centre. As a general rule, a variable quantity in reaching its maximum increases at a less and less rate, and after passing the highest point begins to decrease by insensible degrees. The maximum may indeed be defined as that point at which the increase or decrease is null. Hence it will usually be the most indefinite point, and if we can accurately measure the phenomenon we shall best determine the place of the maximum by determining points on either side at which the ordinates are equal. There is moreover this advantage in the method that several points may be determined with the corresponding ones on the other side, and the mean of the whole taken as the true place of the maximum. But this method entirely depends upon the existence of symmetry in the curve, so that of two equal ordinates one shall be as far on one side of the maximum as the other is on the other side. The method fails when other laws of variation prevail.

In tidal observations great difficulty is encountered in fixing the moment of high water, because the rate at which the water is then rising or falling, is almost imperceptible. Whewell proposed, therefore, to note the time at which the water passes a fixed point somewhat below the maximum both in rising and falling, and take the mean

time as that of high water. But this mode of proceeding unfortunately does not give a correct result, because the tide follows different laws in rising and in falling. There is a difficulty again in selecting the highest spring tide, another object of much importance in tidology. Laplace discovered that the tide of the second day preceding the conjunction of the sun and moon is nearly equal to that of the fifth day following; and, believing that the increase and decrease of the tides proceeded in a nearly symmetrical manner, he decided that the highest tide would occur about thirty-six hours after the conjunction, that is half-way between the second day before and the fifth day after.1

This method is also employed in determining the time of passage of the middle or densest point of a stream of meteors. The earth takes two or three days in passing completely through the November stream; but astronomers need for their calculations to have some definite point fixed within a few minutes if possible. When near to the middle they observe the numbers of meteors which come within the sphere of vision in each half hour, or quarter hour, and then, assuming that the law of variation is symmetrical, they select a moment for the passage of the centre equidistant between times of equal frequency.

The eclipses of Jupiter's satellites are not only of great interest as regards the motions of the satellites themselves, but were, and perhaps still are, of use in determining longitudes, because they are events occurring at fixed moments of absolute time, and visible in all parts of the planetary system at the same time, allowance being made. for the interval occupied by the light in travelling. But, as is explained by Herschel,2 the moment of the event is wanting in definiteness, partly because the long cone of Jupiter's shadow is surrounded by a penumbra, and partly because the satellite has itself a sensible disc, and takes time in entering the shadow. Different observers using different telescopes would usually select different moments for that of the eclipse. But the increase of light in the emersion will proceed according to a law the reverse of that observed in the immersion, so that if an observer notes

Airy On Tides and Waves, Encycl. Metrop. pp. 364*-366*. 2 Outlines of Astronomy, 4th edition, § 538.

the time of both events with the same telescope, he will be as much too soon in one observation as he is too late in the other, and the mean moment of the two observations will represent with considerable accuracy the time when the satellite is in the middle of the shadow. Error of judg ment of the observer is thus eliminated, provided that he takes care to act at the emersion as he did at the immersion.

CHAPTER XVII.

THE LAW OF ERROR.

To bring error itself under law might seem beyond human power. He who errs surely diverges from law, and it might be deemed hopeless out of error to draw truth. One of the most remarkable achievements of the human intellect is the establishment of a general theory which not only enables us among discrepant results to approximate to the truth, but to assign the degree of probability which fairly attaches to this conclusion. It would be a mistake indeed to suppose that this law is necessarily the best guide under all circumstances. Every measuring instrument and every form of experiment may have its own special law of error; there may in one instrument be a tendency in one direction and in another in the opposite direction. Every process has its peculiar liabilities to disturbance, and we are never relieved from the necessity of providing against special difficulties. The general Law of Error is the best guide only when we have exhausted all other means of approximation, and still find discrepancies, which are due to unknown causes. We must treat such residual differences in some way or other, since they will occur in all accurate experiments, and as their origin is assumed to be unknown, there is no reason why we should treat them differently in different cases. Accordingly the ultimate Law of Error must be a uniform and general one.

It is perfectly recognised by mathematicians that in each case a special Law of Error may exist, and should be discovered if possible. "Nothing can be more unlikely than that the errors committed in all classes of observa