429

mean of all the readings when the moon was on the me-
ridian 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. 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 gener-
ally 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

eless

> the will

nd if ngle

the

the

two

ap

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 would give very nearly the true daily mean temperature. At a depth of twenty feet even the yearly fluctuations would become nearly effaced, and the thermometer would stand a little above the true mean temperature of the locality. In registering the rise and fall of the tide by a tide-guage, it is desirable to avoid the oscillations arising from surface waves, which is very readily accomplished by placing the float which marks the level of the water 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.

Grant, History of Physical Astronomy, p. 163.

he

nd

es

y

Determination of the Zero point by the Method
of Means.

There are a number of important observations in which one of the chief difficulties consists in defining exactly the zero point from which we are to measure. We can point a telescope with great precision to a star and can measure the angle through which the telescope is raised or lowered to a second of arc; but all this precision will be useless unless we can know exactly where the centre point of the heavens is from which we measure, or, what comes to the same thing, the horizontal line 90° distant from it. Since the true horizon has reference to the figure of the earth at the place of observation, we can only determine it by the direction of gravity, as marked either by the plumb-line or the surface of a liquid. The question resolves itself then into the most accurate mode of observing the direction of gravity, and as the plumb-line has long been found hopelessly inaccurate, astronomers generally employ the surface of mercury in repose as the criterion of horizontality. They ingeniously observe the direction of the surface by making a star the index. From the Laws of Reflection it follows that the angle between the direct ray from a star and that reflected from a surface of mercury will be exactly double the angle between the surface and the direct ray from the star. Hence the horizontal or zero point is the mean between the apparent place of any star or other very distant object and its reflection in mercury.

A plumb-line is perpendicular, or a liquid surface is horizontal only in an approximate sense; for any irregularity of the surface of the earth, a mountain, or even a house must cause some deviation by its attracting power. To detect such deviation might seem very difficult, because every other plumb-line or liquid surface would be equally

affected by the very principles of gravity. Nevertheless it can be detected; for if we place one plumb-line to the north of a mountain, and another to the south, they will be about equally deflected in opposite directions, and if by observations on the same star we can measure the angle between the plumb-lines, half the inclination will be the deviation of either, after allowance has been made for the inclination due to the difference of latitude of the two places of observation. By this mode of observation applied to the mountain Schehallien the deviation of the plumb-line was accurately measured by Maskelyne, and thus a comparison instituted between the attractive forces of the mountain and the whole globe, which led to a very probable estimate of the earth's average density.

In some cases it is actually better to determine the zero point by the average of equally diverging quantities than by direct observations. Thus in delicate weighings by a chemical balance it is requisite to ascertain exactly the point at which the beam comes to rest, and when standard weights are being compared the position of the beam is ascertained by a carefully divided scale viewed through a microscope. But when the beam is just coming to rest, friction, small impediments or other accidental causes may readily obstruct it, because it is near the point at which the force of stability becomes infinitely small. Hence it is found better to let the beam vibrate and observe the terminal points of the vibrations. The mean between two extreme points will nearly indicate the position of rest. Friction and the resistance of air tend to reduce the vibrations, so that this mean will be erroneous by half the amount of this effect during a half vibration. But by taking several observations we may determine this retardation and allow for it. Thus if a, b, c be the terminal points of three excursions of the beam from the zero of the scale, then (a+b) will be about as much

rection as (b+c) in the other, so

read л these two means, or what is the same,

will be exceedingly near to the point of A sail closer approximation may be made by gareadings and reducing them by the formula

The accuracy of Baily's experiments, directed to detere the density of the earth, entirely depended upon this cde of observing oscillations. The balls whose gravitaden was measured were so delicately suspended by a tetson balance that they never came to rest. The exneme points of the oscillations were observed both when de 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 Jet hand gave double the amount of the deflection.

A most beautiful instance of the mode of avoiding the use of a zero point is to be found in Mr. E. J. Stone's observations on the radiated heat of the fixed stars. The great difficulty in these observations arose from the comParavely great amounts of heat which were sent into the c.cope from the atmosphere, and which were sufficient Oot entirely to disguise the feeble heat rays of a star. .. Mr. Stone fixed at the focus of his telescope a double o electric pile of which the two parts were reversed . Now any disturbance of temperature which upon both piles uniformly produced no effect

galvanometer needle, and when the rays of the were made to fall alternately upon one pile and he total amount of the deflection represented eg power of the star. Thus Mr. Stone

ve with much certainty a heating effect kes, which even when concentrated

2

ounted only to th of a degree

100

PN Novutitie Memoirs,' vol. ii. p. 43, &c.

Fahrenheit, and which represents a heating effect of the direct ray of only about o°00000137 Fahrenheit, equivalent to the heat which would be received from a threeinch cubic vessel full of boiling water at the distance of 400 yards P. It is probable that 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 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 insensibly small. Hence it will usually be the most indefinite point in the whole course, 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

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