ART. XXI.-Theory of Transit Corrections; by ENOCH F. BURK.

AN accurate determination of the element of time is so essential to the purposes of astronomy, that whatever may serve to to simplify or illustrate the modes of obtaining it, may justly be deemed of importance.

In expounding the theory of the transit instrument, the reasoning necessarily partakes much of a metaphysical character. From this cause, combined with limited space, a specific object or the necessities of a popular exhibition, arise certain assumed principles and partial demonstrations in our ablest treatises on the subject. This fact gives occasion for a few supplementary reasonings on the principles of the transit corrections.

There are four prominent corrections to be made in the applications of the transit instrument. These are for errors of observation, inequality of the intervals between the wires, the time of passing over these intervals, error of collimation and deviation of the optical axis from the plane of the meridian. A few suggestions will be made with reference to these corrections in their order.

An observer is liable to error in estimating the instant at which a star transits the wire which indicates the plane of the meridian. To correct for this error, and reduce its probable amount, other wires are introduced into the focus of the instrument, and a mean taken of the times of the transits over them all; and the reduction is proportional to the number of wires introduced.

probability that e is less than when all the errors denoted by e'

n

are of the same sign, and one expression in this case has no advantage over the other. But these errors probably have not all the same sign; but some are negative while others are positive, and thus tend to cancel each other. This constitutes an advantage of the latter expression and renders it probably less than the first. The diminution of error by taking a mean, is probably in pro

e

portion to the number of observations. Let the mean error

n

= the mean error when

the number of observations is increased by unity. Then

is probably less than For if the numerator of the first ex

e n

pression is not probably greater than that of the second while its denominator is greater than the denominator of the other, then the first expression is probably the smallest. But there is no probability that the numerator of the first is greater than that of the second, since there is as much chance that e will be negative as positive.

As the probable reduction of the error of observation is in proportion to the number of wires, it becomes important to connect as many with the instrument as possible. All the advantage, however, of an additional wire to the transit, in the use of the common method of taking a mean, may be gained by adding the times at all the wires to the time at the middle wire and dividing by the number of wires increased by one. This method gives a result probably as accurate as that which would be given by the other with a new wire added to the instrument. Let e= error at the middle wire and e'= sum of errors at all the wires.

ete'

Then according to the method just stated, is the mean er

n+1

ror: a result just the equivalent of that given by the common method with an additional wire, inasmuch as there is no probability that the error at an additional wire will be more favorable in respect to amount and sign to the diminution of the numerator of the fraction expressing the mean, than the error at the middle wire. Indeed, the error at the middle wire is probably less than at an extreme one.

When the intervals between the wires of the transit are unequal, a correction must be applied for this inequality. It is given by eminent authority, that this correction is most perfectly applied by reducing the time at each of the wires separately to the middie wire and then taking a mean, thus requiring as many separate reductions as there are wires less one. But this process is tedious, and may be avoided with advantage in the following manner. Find the place of the mean of the wires. The product of the equatorial distance of this from the middle wire into the secant of the declination, applied with its proper sign to the time of the transit over the mean of the wires, gives the time of the transit over the middle wire. The time of the passage over the mean is found by dividing the sum of the times of the transits by the number of wires, and found with as much probable accuracy as would be connected with the time over the middle wire, obtained in the usual manner, when the intervals are equal. Let y true time at the mean of the wires.

=

d, d,, d, etc. equatorial distances of the wires from the mean. errors of observation at the wires. declination of the star.

e, ev

etc.

Then the observed times for a transit with five wires will be

Now it is evident that if y is taken to represent the time of the transit over the middle wire, and the equatorial intervals be equal, the sum of the coefficients of sec 8 is zero, and there is a liability to just as large an error of observation at each wire. Hence the

E

time of transit at the middle wire would be found 7(2): a result no more accurate for the middle wire than was the previous one for the mean of the wires.

If to expression (1), the product of the equatorial distance of the mean of the wires from the middle wire be applied with the proper sign, y, which represents the true time of passage over the mean of the wires, will become the true time of passage over the middle wire, and the expression (1) will become identical with expression (2): the same result will be obtained as if the equatorial intervals had all been equal.

The correction for the inequality of the intervals may also be made, by reducing the times of the transits on one side of the middle wire, to what they would have been, if each wire had been as far from the middle as is the one corresponding to it on the other side, and then using these reducing times in obtaining

a mean.

Let y= true time of passage at the middle wire d', d", etc.= true times between the middle and other wires.

will be the observed times of the transits. Now conceive a quantity badded to d", such that d" ± b=d"", and also another quantity b'added to d', such that d'b'd""". In this case, when the quantities are added, the second column will disappear,

E

and dividing by 5, we have 7 as accurate a result as is obtained when the intervals between the wires are all equal; since the observer is liable to just the same errors at the wires.

It is well known that the time occupied by a star in passing over a given part of the field of view, is not accurately expressed by the product of its equatorial value into the secant of the declination. This arises from the inequality of the arcs of different diurnal circles, intercepted between the same limits. The formula which is strictly accurate, is, in its common form,

Where y time of passage from one wire to another, equatorial interval between them.

But there is another known form, which, while it indicates the operations to be performed more clearly than the last, is sometimes otherwise more convenient for use. It may be thus de

monstrated.

Let x arc of a diurnal circle intercepted between two wires. y = arc of equator intercepted between the same.

x a part of x equal to y.

Then y = y. sec 8.

yx': yx::x:x

=ry.sec +37.5. arc3 1". yy3. sec3 8.

The two corrections of the time for error of collimation and deviation of the transit axis from the plane of the prime vertical, are given respectively by the expressions c. sec. and a. sin (-8). sec ; so that the equated error of the clock becomes E--a.sin (-8). sec &-c. sec 8. In these expressions is the latitude of the place of observation, Æ the right ascension of a star, and a, c, the deviation and error of collimation. It is plain that the values of a and c might be determined from any three equations of the error of the clock, but this method is much less accurate, on account of errors of observation, than another which is sometimes employed. This latter method may be ex

emplified with reference to c, and consists in subtracting two equations, one of which is the sum of several such equations of the error of the clock as contain large coefficients of c, and the other the sum of an equal number of such equations as contain small coefficients of c, and such coefficients of a that their sum will nearly balance and cancel the other coefficients of a in the subtraction. Then the error of the clock disappears, the term into which a enters being very small, may be neglected, and c becomes known. The superior accuracy of this method, which we find assumed, it may be well to establish.

Before proceeding farther, however, it should be observed that while the expression for the effect of the error of collimation upon the time, needs in strictness of theory to be subjected to the same modification as that which commonly expresses the time of a star between two wires, the difference between the value of this expression and the true correction, is always practically inappreciable. The time occupied by a star in passing from a small circle of the sphere parallel to the meridian, to the meridian, is equal to the space passed over, expressed in time, into the secant of the declination; but this space or the arc of the diurnal circle intercepted between the two circles, is different for different declinations, and hence the constant c can only represent it approximately. In practice, however, the difference is of no consequence. Let 0"-05 be the greatest allowable error in the value of the correction arising from the use of the form c. sec d. Then at 88° 30', the declination of Polaris, the error of collimation must be 105; which is greater than it will ever be when any sort of care is used in making the adjustment.

The value of c as found by the common method, is much less affected by errors of observation than that found from any three equations. The effect of these errors on the value of c as determined by the common method, is expressed by the algebraic difference of two sets of errors into a coefficient which is a very small fraction, and which may be made of any degree of smallness, while the effect of the errors of observation on c as found by the other method, is expressed by two terms each of which is the algebraic difference of two single errors into a coefficient which is probably integral. Now, since there is no probability that the sum of these two errors is less than the greatest, if it can be shown that the coefficient of either is integral, and that there is no likelihood that its factor is any less than the corresponding factor of the term which expresses the effect of the errors of observation in the common method, it follows that this effect is probably much less than that incident to the other method.