TO CONVERT MEAN TIME INTO SIDEREAL TIME, AND SIDEREAL INTO MEAN TIME. 24. Since 365 solar days = 3664 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 1 365.25 may be changed to mean time by diminishing it by its ment, its 1 366.25 part. The first part of the table gives, for each 10 minutes of the argu1 part, by which it is to be increased. The second. 365.25 part of the table gives the 1 366.25 part of the argument. 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 time. Given mean time, 17h 48m 365.70 Corr. for 17h 40m, Ex. 2. time. 2m 548.13 To change this interval of sidereal time back to mean 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 3 66 66 66 0.477 12, we are not justified in drawing any conclusion whatever respecting the logarithms of numbers between 2 and 3. Hence some one or more hypotheses are always necessary as the base of any system of 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 ' 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. THEOREM. The differences of the sum of two quantities are equal to the sums of their differences. f1 fa fa, etc., be one set of functions; f, f, f', etc., another set. ƒ1 + fi', ƒ2 +ƒ‚'‚ ƒ, +ƒ‚', etc., will then be their sums. In the first of the following columns we place the first differences off, in the second those of f', and in the third those of.ƒ + ƒ', each formed according to the rule: It will be seen that the quantities in the third column are the sums of those in the first two. We see that the third set of values of 4' follow the theorem. Because the second differences are the differences of the first, the third the differences of the second, etc., it follows that the theorem is true for differences of any order. Now when we write a series of functions in which the decimals exceeding a certain order are omitted, we may conceive each written number to be composed of the algebraic sum of two quantities, namely: 1. The true mathematical value of the function. 2. The negative of the omitted decimals. Example. In the preceding collection of logarithms, since the true value of log 30 is 1.477 121 3..., we may conceive the quantity written to be 1.477 12 = log 30.000 001 3.... Hence the differences actually written are the differences of the true logarithms minus the differences of the errors. Now suppose the errors to be alternately + 0.5 and 0.5 the point marking off the last decimal. Their differences will then be as follows: It is evident that the nth order of differences of the errors are equal to 2n-1. Hence, in this case, if the nth order of differences of the true values of the function were zero, still, in consequence of the omission of decimals, the actual differences of the nth order would be 2n-1. This, however, is a very extreme case, since it is beyond all probability that the errors should alternate in this way. A more probable average example will be obtained by supposing a single number to have an error of 0.5, while the others are correct. We shall then have: Aiv + 0.5 2.0 1.5 1.0 0.5 0 0 0 + 0.5 +1.5 + 3.0 5.0 2.0 0.5 + 0.5 + 2.5 + 5.0 In this case the maximum value of the difference of the nth order is 1.5 in the differences of the third order, 3 in those of the fourth, 5 in those of the fifth, etc. Its general expression is This being about the average case, in actual practice the differences may be two or three times as great without necessarily implying an error greater than 0.5 in the numbers written. We have now the following general rule for judging whether a series of numbers do really follow a uniform law: Difference the series until we reach an order of differences in which the and signs either alternate or follow each other irregularly. - |