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respectively, and obtain in the same way 7 + 17a + 2160. Solving, we find a = -1.4, b = .8, and substituting in our original equation 0=1+ ax + by, we have 0 = 1 - 1.4x + .8y, or y 1.75x1.25. Constructing the line thus found, we obtain MN, Fig. 1, which will be seen to agree very well with our original conditions.

For a fuller description of the various applications of the Theory of Probabilities to the discussion of observations, the reader is referred to the following works. Méthode des Moindres Carrées par Ch. Fr. Gauss, trad. par J. Bertrand, Paris, 1855, Watson's Astronomy, 360, Chauvenet's Astronomy, II, 500, and Todhunter's History of the Theory of Probabilities. A good brief description is given in Davies' and Peck's Math. Dict., 454, 536 and 590, also in Mayer's Lecture Notes on Physics, 29.

Peirce's Criterion. It has already been stated that all observations affected by errors not accidental, or mistakes, should be at once rejected. But it is generally difficult to detect them, and hence various Criteria have been suggested to enable us to decide whether to reject an observation which appears to differ considerably from the rest. One of the best known of these is Peirce's Criterion, which may be defined as follows:-The proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection, multiplied by the probability of making so many and no more abnormal observations. Or, to put it in a simpler but less accurate form, reject any observations which increase the probable error, allowing for the chances of making so many and no more erroneous measurements. Without this last clause we might reject all but one, when the probable error by the formula would become zero. See Gould's Astron. Journ., 1852, II, 161; IV, 81, 137, 145.

Another criterion has been proposed by Chauvenet, which, though less accurate than the above, is much more easily applied. It is fully described in Watson's Astronomy, 410.

Differences. To determine the law by which a change in any quantity A alters a second quantity B, we frequently measure B when A is allowed to alter continually by equal amounts. Thus in the example of the boiling of water, we measure the pressure

corresponding to temperatures of 0°, 10°, 20°, 30°, &c. Writing these numbers in a table, by placing the various values of A in the first column, those of B in

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from that which follows it, and so on.

Interpolation. One of the most common applications of differences is to determine the value of B for any intermediate value of A. This is done by the formula,

B=Bm+n Dm'+

n(n − 1)D " +

m

1.2

n(n − 1) (n − 2) D_""'+&c,

A. B.

1. 2. 3

m

D'.

D". D"". D''"'.

in which Bm is the measurement next preceding B; Dm, Dm", D", the 1st, 2d, 3d, differences, and n a fraction equal to (A–— Am) ÷ (Am+1 — Am), in which A, Am correspond to B, Bm, and Am+1 is the next term of the series to Am The use of this formula is best shown by an example. Suppose, from the accompanying table, we wish to find the value of B corresponding to A 12.5. We have Вm 1728, Dm

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10 1000

+331

11 1331

+66

+397

+6

12 1728

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+6

13 2197

+78

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+6

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+84

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+631

= 12.5

+6

15 3375

+90

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12, Am+1 13, A =

and n = (12.5 12) (13

-12).5. Hence,

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9.75 + .375 + 0 = 1953.125.

In this particular case B is always the cube of A, and it may be

seen that our formula gives an exact result.

the 4th, and all following differences, equal zero.

The reason is that

Inverse Interpolation. Next suppose that in the above example we desired the value of A for some given value of B, as B'; that is, in the equation,

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we wish to find n. Evidently it is impossible to determine this exactly, but an approximate value may be found by the method of successive corrections. Neglect all terms after the third, and deduce n from the equation,

B' = B„ + nD„' + n(n − 1)D_",

1.2

m

which is a simple quadratic equation. Substitute this value of n in the terms we have neglected, and call the result N, then

B' = Bm + nD +

m

n(n · 1)
1.2

D" + N,

from which again we may deduce a more accurate valne of n. This again gives a new value of N, and by continuing this process we finally deduce n with any required degree of accuracy.

It is sometimes more convenient to neglect the third term, and deduce n from the equation B' = Bm + nDm', which saves solving a quadratic equation, but requires more approximations. The values of n(n - 1) ÷ 1.2, n(n − 1) (n − 2) 1. 2. 3, &c., may ÷ be more readily obtained from Interpolation tables than by computation. A good explanation of this subject is given in the Assurance Magazine, XI, 61, XI, 301, and XII, 136, by Woolhouse.

2

When the terms are not equidistant the method of interpolation by differences cannot be applied. In this case, if we wish to find values of B corresponding to known values of A, we assume the equation, B = a + bA+ cA2 + d43+ &c., and see what values. of a, b, c, &c., will best satisfy these equations. If we have a great many corresponding values of A and B, the method of least squares should be applied. In general, however, it is much more convenient to solve this problem by the Graphical Method de

scribed below. See Cauchy's Calculus, I, 513, and an article in the Connaissance des Temps, for 1852, by Villarceau.

Numerical Computation. Where much arithmetical work is necessary to reduce a series of observations, a great saving of time is effected by making the computation in a systematic form. In general, measurements of the same quantity should be written in a column, one below the other, instead of on the same line, and plenty of room should always be allowed on the paper. When the same computations must be made for several values of one of the variables, instead of completing one before beginning the next, it is better to carry all on together, as in the following example. Suppose, as in the experiment of the Universal Joint, we wish to compute the values of b in the formula, tan b = cos A tan a, in which A 45°, and a in turn 5°, 10°, 15°, &c. Con

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In the first line write the various values of a, in the second the corresponding values of its log tan, and so on throughout the computation. An error is purposely committed in the above table to show how easily it may be detected. It will be noticed that the values of b increase pretty regularly, except that when a = 25°, and that this is but little greater than that corresponding to a = 20°. Following the column up we find that the same is the case for log tan b but not for log tan a, hence the error is between the two. In fact, in the addition of the logarithms we took 6 and 4 equal to 10, and omitted to carry the 1; log tan b then really equals 9.51815, and b = 18° 15′. If the error is not found at once this value of b should be recomputed. Besides these advantages, this method is much quicker and less laborious. When we have to multiply, or divide by, the same number A a great many times, it is often shorter to obtain at once 14, 2A, 3A, 4A, &c., and use these numbers instead of making the multiplication each time. This is useful in reducing metres to inches, &c. There are many

other arithmetical devices, but their consideration would lead us too far from our subject.

Significant Figures. One of the most common mistakes in reducing observations is to retain more decimal places than the experiment warrants. For instance, suppose we are measuring a distance with a scale of millimetres, and dividing them into tenths by the eye, we find it 32.7 mm. Now to reduce it to inches we have 1 metre = 39.37 in., hence 32.7 mm. = 1.287399. But it is absurd to retain the last three figures, since in our original measurement, as we only read to tenths of a millimetre, we are always liable to an error of one half this amount, or .002 of an inch. Then we merely know that our distance lies between 1.2894 and 1.2854 inches, showing that even the thousandths are doubtful. It is worse than useless to retain more figures, since they might mislead a reader by making him think greater accuracy of measurement had been attained.

If we are sure that our errors do not exceed one per cent. of the quantity measured, we say that we have two significant figures, if one tenth of a per cent. three, if one hundredth, four. Thus in the example given above, if we are sure the distance is nearer 32.7 than 32.8 or 32.6, we have three significant figures, and it would be the same if the number was 327,000, or .00327. In general, count the figures, after cutting off the zeros at either end, unless they are obtained by the measurement; thus 300,000 has three significant figures if we know that it is more correct than 301,000 or 299,000. In reducing results we should never retain but one more significant figure than has been obtained in the first measurement, and must remember that the last of these figures is sometimes liable to an error of several units.

Successive Approximations. This method is also known as that of trial and error. It consists in assuming an approximate value of the magnitude to be constructed, measuring the error, correcting by this amount as nearly as we can, measuring again, and so on, until the error is too small to do any harm. As an example, suppose we wish to cut a plate of brass so that its weight shall be precisely 100 grammes. We first cut a piece somewhat too large, weigh it and measure its area. If its thickness and density were perfectly uniform we could at once, by the rule of

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