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coefficients lie between the limits o"or and o"oo. I take this lower limit as that beyond which it is unnecessary to go. In estimating the S.D. I take oor as the unit. An approximate estimate of the distribution of these coefficients in magnitude shows that the square of the S.D. produced by omitting them is, with sufficient exactness,
€2 = 28.
There remain, by a rough estimate, about 300 terms exceeding the limit o" or, nearly all of which I assume will be tabulated. I also assume that the individual numbers of these tables will be formed by carrying each number to o"oor, and that the last decimal will be dropped in the final tabulated number, which will be given only to o'o. Making due allowance for all imperfections, I find that the standard deviation of a number interpolated from a single table thus formed will be 026 if the number is written down to one decimal beyond that of the tables. But if the number is, as usual, only written to the tabular number of decimals, the deviation will be 0 39.
I assume that not more than 120 tabular numbers will be added to form the longitude. The contribution to (S.D.)2 arising from the summation of these numbers is
120 X 0.392 = 18.2.
The imperfections of the arguments will also have their influence. The deviation produced by them may be reduced by giving each argument to one decimal beyond that required by the condition that the error arising from a unit error of the argument shall always be less than that of the tabular unit. But, without going beyond this rule, the effect of the errors of the argument will not exceed that arising from the addition of thirty more tabular numbers. We may thus have—
(S.D.)2 from errors of arguments = 4'5.
In some cases the effect of the imperfections of the correction for the second differences may add to the S.D. I think that, with a little skill and attention on the part of the computer, this S.D. need not exceed ±02, giving 48 for its entire contribution to (S.D.)2.
Summing up all the sources of accidental deviation of the tabular results from theory we have
We shall therefore have, in the case of each computed longitude, a S.D. of ±0"075 and a probable error of o"05.
§4. Degree of precision required.
Let us compare this with the degree of precision required in a comparison with observation. My experience in the use and examination of lunar observations leads me to the conclusion that no single observation of any sort can be practically made without a mean error exceeding to"5, corresponding to a probable error of ±0" 34. It is desirable that the mean error of a co-ordinate found from the tables should be less than this. But all the results obtained from lunar observations depend upon a great number of observations, which make the accidental errors unimportant in comparison with the systematic ones. The practical advantage of a degree of precision above that just mentioned in the tabular places is very small, and is practically evanescent if reduced below to"4. This degree of precision will be surpassed by adopting as the unit to be tabulated the 10-8 part of the circumference, or oo1296. Of course the unit o"or would answer our purpose if deemed more convenient in use. But the smaller the numbers used, the easier the computer will find it to avoid small errors, while the S.D. will still fall below to"09.
There is, however, one point still to be considered in this connection which may modify our conclusions. Granting an S.D. of too in the individual longitudes, we must expect that in the course of a year there may be three or four of the 730 tabulated longitudes in error by three times the S.D., and possibly one of four times this amount. But every error approaching such a magnitude as this will be detected by differencing the 12-hour ephemeris. A legitimate proceeding will then be to smooth off the ephemeris by such small corrections as shall reduce the higher, say the fifth or sixth, differences to a sufficiently smooth series. Each corrected tabular result may then be regarded as the mean of two or more neighbouring quantities, and the maximum error of the ephemeris will be reduced nearly to the mean S.D. In a word, we may fairly count on having an ephemeris in which all the errors exceeding some limit between o"12 and o" 30 will be eliminated. This limit is still within the errors of the best observations, and the cases in which it is approached will be rare.
§ 5. Reduction to sexagesimal units.
The proposed units will require the reduction of the final longitudes to degrees, minutes, and seconds. The tables necessary for this purpose will perhaps fill four pages, and the computation will be equivalent to the entry of three additional tables.
If the unit oor is deemed preferable, its use will still require some study. It was adopted in Peirce's Tables of the Moon,
published in 1853. The tabular numbers were there expressed in degrees and seconds, minutes being ignored. I found the use of this system cumbrous, and should prefer to use seconds pure and simple, subtracting 1,296,000" or its multiples when necessary. The proposed circumferential unit does away with this subtraction. Although it is a little easier to change the degrees into seconds and minutes than it is to change the circumferential unit, I still think the advantage to lie with the latter.
It may be of interest in this connection to note that if we should base the unit on the degree, tabulating to o° 00001, the S.D. of the individual longitudes would still be only o"18, and we might be fairly confident that no error exceeding o" 4 would remain in a smoothed-off annual ephemeris as often as once a year.
Of course all this presupposes that the computer is always careful never to make a greater error than o‘5 in interpolating and writing down his number. The question may arise whether it is not well to allow him a margin of one or two units, by adopting smaller units. My answer is that the labour of handling large numbers involves more mental strain than that required in the accurate handling of small numbers, and that the assigned standard of precision will be more easily reached by the careful use of the smaller numbers than by the careless use of the larger ones.
§ 6. Epochs and Arguments.
For the practical work of computing places of the Moon for given dates, I do not think that any system more convenient than the usual one can be devised. The Hansenian form, in which the Gregorian and Julian calendars are used, is the most convenient of all. But it is always desirable to give the tables such a form that the relation between the tabular numbers and the original elements shall be easily examined, and corrections to the theory readily applied. This suggests a slight sacrifice of ease in computing an isolated position, or an ephemeris, to the requirements of the theoretical investigator.
Simplicity in the other direction is reached by the use of days of the Julian period. This was first employed, I believe, by Peirce, and is now extensively used in astronomy, especially in Oppolzer's works relating to eclipses. In using this system, a first and easy step is the reduction of the ordinary calendar date to days of the Julian period. Then absolute uniformity is reached in the construction and use of the tables.
The principal immediate drawback of this system is that, if used unmodified, the period of 1000 days must take the place of the year. The formation of the arguments of short period is then inconvenient. Many of the lunar arguments have periods not differing much from a month. From 12 to 15 multiples then suffice when the year is used, but with the period of 1000 days the number of multiples to be tabulated and subtracted will frequently be between 30 and 40, and some times more. Of course
this difficulty can be lessened by taking 500 days instead of 1000 as the second unit. But this will detract from simplicity of form.
A yet more serious drawback to the theoretical investigator is that the fundamental epochs usually adopted in astronomy, and for which the elements must be found, do not correspond to any power of 10 in the days of the Julian period. A complete transformation of the elements is therefore required to form the numbers on which the tables are based. If, therefore, multiples of 500 or 1000 days are used instead of years, I should prefer to count them back from 1900'o, thus gaining all the advantages of the Julian period without any other disadvantage than that of non-correspondence with the eclipse and other tables of Oppolzer.
The reduction of such a system to the ordinary calendar may be made a very simple matter. It seems to me, therefore, that the maximum of advantage will be reached by giving the fundamental arguments for cycles and periods based on multiples of 500 days before and after the fundamental epoch 1900 Jan. o.
Probably the most convenient fundamental quantities to tabulate will be the longitude of the node, and the mean distances of the Moon and of its perigee from the node, all expressed in circumferential units. Then, whatever form the tables may be thrown into, we shall have the nearest approach to a simple, straightahead computation.
Finally, a serious problem is that of summing perhaps 100 periodic terms with coefficients not differing greatly from o"01. Ì have devised a machine for this purpose, the description of which must form the subject of another publication.
An Example of Professor Karl Pearson's Calculation of Correlation in the case of the Periodic Inequalities of Long-period Variables. By H. H. Turner, D.Sc., F.R.S., Savilian Professor.
1. The following note is written with a twofold purpose. Firstly, it is hoped that an indication of some value has been obtained with regard to the features of "long-period" variability; and secondly, the opportunity is taken to write out in full a simple example of the calculation of "correlation" between quantities by the methods of Professor Karl Pearson.
In the M.N. for March last (p. 416) Professor Pearson himself gave au admirable summary of methods; but he naturally did not repeat the elementary working which has become so familiar to him, and has been given often before in other connections. There are doubtless many to whom this working is already familiar; but there are certainly many others who do not know it and who might use it if they had an astronomical example readily accessible. In these busy days many people have not the leisure to search for references in scientific literature outside their own subject.
2. In support of my view that there is need for an example, I may cite an illustration shown me by Professor E. C. Pickering (see Observatory, March 1905, p. 153).
Suppose we have the following observed values of A and B:
First group according to B, in sets of three for which B has
Now, neither Professor Pickering nor myself was prepared at that time to deal with this situation by a definite process; and yet this is just an elementary case of the kind which Professor Pearson's methods were devised to meet.
This example is worked out below, and it is shown that the numerical measure of the correlation is
r = '08±22
or almost nothing at all: so that, in spite of appearances to the contrary, we are not entitled to assume any relationship between A and B. Putting it in another way, one proposed relation (AB) is as good as another (A-7)=2(B-7).
3. A point of detail may be mentioned here. In much statistical work, a large number of figures are used. Thus we get such statements* as
The probable error of r being about 01, some of these figures are superfluous; and in what follows fewer figures are used. This, however, represents a personal view which is, I find, not generally approved by other workers.
4. The particular example selected for treatment is the discussion of the elements of maximum given by Chandler for long-period variables. The following particulars are taken from
Frequency Curves and Correlation, by W. P. Elderton, p. 119: an excellent little book, from which much is to be learned.