Here y is the element required for the given time.

A is the element for the noon, midnight or complete hour preceding the given instant.

m is the given number of hours since noon or midnight, or the number of minutes since the complete hour.

n is either 24 h., 12 h., or 60 m, according to the interval between the times for which the elements are given in the Nautical Almanac.

A', A", &c., are the first, second, &c. differences of the elements in the almanac.

EXAMPLE. Let it be required to find the moon's latitude for August 4th, 1842 at 16 h. 18' mean time at Greenwich, that is, at 4.3 h. after mean midnight.

The latitudes are taken from the Nautical Almanac; the positive sign indicates north latitude; the negative sign, south latitude.

Here A0° 5' 54".6, A'=-40′ 27′′.7, or -40.463. m.

m

= 0.358, A' = 14′ 29′′.16, ▲′′= +11′′.4

n

Therefore y0° 8' 35.87, the moon's correct latitude, south.

318. The horary motions of the sun, moon, and planets, when great accuracy is required, must also be taken from the almanac with an attention to second differences. When the elements are given in the almanac for every 12 hours; the first difference between them at the noon or midnight preceding, and that which follows the given instant being divided by 12, will give the mean horary motion during the 12 hours: this must be corrected by applying to it with its proper sign the difference between the values of the third term in the formula for interpolation for the complete hour before, and for the complete hour after the given instant; for this difference will express the quantity by which the true differs from the mean hourly motion at the given time. Thus, wanting

the moon's hourly motion in latitude at the time given in the above example, we have

=-3′ 22′′.31 for the mean hourly motion:

but the value of the third term in the series is,

1

9

35 288

for 4 hours after midnight, = (11".4)=-1".27, for 5 hours =-3(11".4) = 1.39. The difference between these is 0".12, which added to the mean hourly motion, gives 3′ 22′′.43 for the true hourly motion between 16 hours and 17 hours on the given day.

When the moon's horary motion in declination is required for any given time, the mean horary motions must be taken out for at least three complete hours, including within them the given time (they may be obtained by multiplying each of the differences for 10 minutes, corresponding to those hours, by 6), and these may be considered as the horary motions for the middles of the intervals: the differences between them will be nearly constant, but a mean of the differences may be considered as a first difference, which being multiplied by

m

n

will give the variation of the hourly motion; and this must be added to, or subtracted from, the mean hourly motion in order to give the correct hourly motion at the given time. Here n = 60', and m is the number of minutes between the given instant and the middle between the complete hour preceding and following it.

319. The subjoined process for obtaining a series of interpolated numbers by means of second differences will occasionally be found useful. Let the numbers in the column R below be the radii vectores of a comet corresponding to certain given longitudes or anomalies, reckoned on the orbit, and differing from one another, for example, by one degree; and let it be required to interpolate the numbers for every 10

Then between 30° and 32° there will be twelve first dif

ferences, and their sum will be 4355.

Let d be the first of these twelve first differences, and 8 the second difference, supposed to be constant: then the several first differences will be

d, d + 8, d + 28, &c. ... to d+118;

and their sum will be 12d + (1 + 2 + 3 +

....

or 12d + (1 +11) 118: hence 12d + 668 = 4355.

+11) 8,

In like manner the sum of the six first differences between 30° and 31° will be 6d + 158 2142.

From these equations we find d = 352.07, 8 = 1.972: then for 30° we have R = 1.11072;

for 30° 10',

for 30° 20′,

for 30° 30',

RR+ d = 1.11424;

R" R'+ d + 8 = 1.11778;

R=R" + d + 28 = 1.12134; &c.

320. In the simple circumstance of observing a signal, as the flash of gunpowder or the occurrence of a celestial phenomenon, the estimates made by two persons of the instant at which the event took place will, in general, differ; and the same person does not, on different occasions, observe phenomena, with respect to time, in like manner: in the bisection of a star by a wire in a transit telescope, such difference has amounted to several tenths of a second. The incongruity is conceived to arise from peculiarities in the organs of vision, or in the perceptions of different individuals, and from variations in the state of the nervous system, according, probably, as the person may be more or less fatigued at the time of making the observation.

321. The correction which should be applied, on account of this cause of error, to the observed time of the occurrence of a phenomenon is called the personal equation; and, in the present state of Astronomy, it is incumbent on every observer, by comparing the results of his observations with those deduced from the observations of other persons, to determine the value of the correction which should be applied. At the Greenwich Observatory, when a difference exists between the personal equations of two observers, the parties determine the error of the sidereal clock at a certain instant, from the transits of stars observed by them on alternate days; and the difference between the errors is considered as a correction to be subtracted from the several errors determined by the observer whose personal equation is the greatest, in order to reduce them to the values which would have been determined from the observations made by the other. In general, when a terrestrial signal or a celestial phenomenon is to be observed by different persons simultaneously, at every repetition of the observation the parties should interchange their stations, in order that the equations may be detected and determined.

322. It is customary for observers to estimate the goodness of a single observation according to a judgment formed at the

time under existing circumstances; and when they have made two or more observations which are presumed to differ in degrees of goodness, numbers expressing the relative values are multiplied into the numerical expressions of the observations, in order that all the observations may be reduced to the same standard with respect to correctness. Thus, if one observation should appear to merit confidence twice as much as another, it would be multiplied by 2; and, in general, A1, and A2, representing the numerical expressions for two observations whose relative merits are denoted by two numbers represented by w, and w2, respectively, the two observations, when reduced to one standard, would be expressed by w1 A1 and w2 A2. The numbers wi &c. are called weights; and each, when applied to an observation, may be conceived to represent the number of observations of standard goodness to which the observation is equivalent. The weight due to several observations taken collectively is equal to the sum of the weights due to the observations separately: thus W1A1 + W2 A 2 + &c. is taken to express the weighted mean

1

W29

of the observations A1, A2, &c. collectively.

323. If the analytical expression given by writers on Probabilities (De Morgan, Theory of Probabilities, in the Encyclop. Metropol., art. 100.) for the probability that the error in a single observation lies within certain limits expressed by and e(e representing some small number) be made equal to, which denotes equality of chance that an observed result may differ from the truth in excess and defect by equal quantities; and if from the equation so formed the value of e be obtained, it will be found to be equal to 0.476936. Let this be represented by ɛ, and if it be substituted in the expression for the limit of the probability of error, on taking the number representing the common average or arithmetical mean of several observations as the true mean (art. 115. in the work above quoted), that expression becomes, denoting the sum of several terms of the like kind by one such term having prefixed to it the symbol Σ,

or, which is proved to be an equivalent expression,

1

here Σ. E2 (representing E2,+E22+ &c.) is the sum of the squares of the presumed errors in the several observations, or of

the differences between the actual observations and the average

Σ.A2

n

or arithmetical mean of all; represents what is called the mean square of the numerical expressions for the several observations A1, A2, &c., or the sum of the squares of the separate observations divided by n, the number of observations, and (2.4) 3

2

is the square of the arithmetical mean of all the

observations: in the formulæ, ɛ √/2=0.67449.

324. It is evident that the precision of the result obtained by taking the ordinary average of several observations as a true result, will be so much greater as either of the expressions (A) or (B) is smaller; or, using the first expression, the

degree of precision is inversely proportional to

n

directly proportional to E.E2), that is, to

n

1

n

n2

Σ.Ε, or

E. E2° It fol

lows also, from the expression (A) that in two sets consisting of an equal number of observations, the precision of the results is inversely proportional to E.E2; and if the mean

Σ.Ε

n

square of the errors, be the same in both, which implies that all the observations are equally good, the precision is directly proportional ton. Thus, if there be taken the ordinary average of two sets of observations unequal in number, but all equally good; in order to reduce the averages to the same degree of precision, each must be multiplied by the square root of the number of observations from which it was obtained.

n

325. The term Σ.E2 or

n

n2 Σ.E2

is considered as represent

ing the weight due to the average of a series of observations ; therefore the precision due to the result of a series of observations varies with the square root of the weight. From this formula for the weight an observer may learn to determine the weight due to any single observation made by himself: for, having obtained the numerical values of several observations (for example, the particular number of seconds indicated by a clock at the bisection of a star by a wire in a telescope), and taken the difference between each and an arithmetical mean of all, let him consider the several differences as so many errors; then, on dividing the square of the number of observations by the sum of the squares of the errors, the result