TABLE IX.

TO CONVERT TIME INTO ARC, AND VICE VERSA.

23. In astronomy the right ascensions of the heavenly bodies are commonly given in hours, minutes, and seconds, the circumference being divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.

8

18 = 15";

the signs 1, ", and indicating hours, minutes, and seconds of time. Hence we may change time into arc by multiplying by 15, and arc into time by dividing by 15, the denominations being changed in each case. Table IX. enables us to do this by simple addition and subtraction by a process similar to that employed in changing hours, minutes, and seconds into decimals of a day.

To turn time into arc, we find in the table the whole number of degrees contained in the time denomination next smaller than the given one, and subtract the former time denomination from the latter.

Next we find the minutes of arc corresponding to the given time next smaller than the remainder, and again subtract.

Lastly we interpolate the seconds corresponding to the second remainder.

The computer should be able to go through this operation without writing down anything but the result.

The operation of changing arc into time is too simple to require description, but it is more necessary to write down the work.

EXERCISES.

Change the following times to arc, and then check the results by changing the arcs into time and seeing whether the original times are reproduced:

TABLE X.

TO CONVERT MEAN TIME INTO SIDEREAL TIME, AND SIDEREAL INTO MEAN TIME.

1 365.25

1 part, by which it is to be increased. The second 365.25 part of the table gives the

may be changed to mean time by diminishing it by its part. The first part of the table gives, for each 10 minutes of the argument, its

=

24. Since 365 solar days 366 sidereal days (very nearly), any period expressed in mean time may be changed to sidereal time by increasing it by its part, and an interval of sidereal time

1 366.25

part of the argument.

1 366.25

The small table in the margin shows the change for periods of less than 10 minutes.

Example 1. To change 17h 48m 36.7 of mean time to sidereal

Ex. 2. To change this interval of sidereal time back to mean

Change to sidereal time:

1. 3h 42m 36.5 m. t.; 3. 22h 3m 5.61 m. t.

2. 18h 46m 29.82

Change to mean time:

66

4. Oh 1m 128.55

66

OF DIFFERENCES AND INTERPOLATION.*

25. General Principles.

We call to mind that the object of a mathematical table is to enable one to find the value of a function corresponding to any value whatever of the variable argument. Since it is impossible to tabulate the function for all values of the argument, we have to construct the table for certain special values only, which values are generally equidistant. For example, in the tables of sines and cosines in the present work the values of the functions are given for values of the argument differing from each other by one minute.

The process of finding the values of functions corresponding to values of the argument intermediate between those given is called interpolation.

We have already had numerous examples of interpolation in its most simple form; we have now to consider the subject in a more general and extended way.

In the first place, we remark that, in strictness, no process of interpolation can be applicable to all cases whatever. From the mere facts that

To the number 2 corresponds the logarithm 0.301 03,

66 66

66

66 3

66

66 0.477 12,

Hence some one or base of any system of

we are not justified in drawing any conclusion whatever respecting the logarithms of numbers between 2 and 3. more hypotheses are always necessary as the interpolation. The hypotheses always adopted are these two:

1. That, supposing the argument to vary uniformly, the function varies according to some regular law.

2. That this law may be learned from the values of the function given in the table.

These hypotheses are applied in the process of differencing, the

* The study of this subject will be facilitated by first mastering so much of it as is contained in the author's College Algebra, §§ 299–302.

It is also recommended to the beginner in the subject that, before going over the algebraic developments, he practise the methods of computation according to the rules and formulæ, so as to have a clear practical understanding of the notation. He can then more readily work out the developments.

nature of which will be seen by the following example, where it is applied to the logarithms of the numbers from 30 to 37:

The column 4' gives each difference between two consecutive values of the function, formed by subtracting each number from that next following. These differences are called first differences.

The column 4" gives the difference between each two consecutive first differences. These are called second differences.

In like manner the numbers in the succeeding columns, when written, are called third differences, fourth differences, etc.

Now if, in continuing the successive orders of differences, we find them to continually become smaller and smaller, or to converge toward zero, this fact shows that the values of the functions follow a regular law, and the first hypothesis is therefore applicable.

In order to apply interpolation we must then assume that the intermediate values of the function follow the same law. The truth of this assumption must be established in some way before we can interpolate with mathematical rigor, but in practice we may suppose it true in the absence of any reason to the contrary.

26. Effect of errors in the values of the functions. In the preceding example it will be noticed that if we continue the orders of differences beyond the fourth, they will begin to increase and become irregular. This arises from the imperfections of the logarithms, owing to the omission of decimals beyond the fifth, already described in § 11.

When we find the differences to become thus irregular, we must be able to judge whether this irregularity arises from actual errors in the original numbers, which ought to be corrected, or from the small errors necessarily arising from the omission of decimals.

The great advantage of differencing is that any error, however small, in the quantities differenced, unless it follows a regular law, will be detected by the differences. To show the reason of this, we investigate what effect errors in the given functions will have upon the successive orders of differences.