convertible terms-as explained in Vol. I-it means that symmetrical waves of all sizes are represented by Fourier's Series; it is only their varied superposition that brings out the odd, irregular curves so often seen in their resultant— the deviations.

CHAPTER XXIII.

THE MAGNETIC COEFFICIENTS.

Section One: The Method of Least Squares.

304. Necessity for adjustment of the observation-equations.-Eq. (125), for calculating the deviations, contains five unknown quantities—A, B, C, D, E—and if observations be made on only five points, we should then have but five equations similar to (125), and the unknown quantities could be deduced directly by any process of algebra; but we may observe on 8, 16, or 32 points and have a corresponding number of equations, or on each half point and have 64-or even on every degree of the circle and have 360 equations: but in all such cases we have only five unknown quantities to be determined. When their values are ascertained from observations on only five points, such values satisfy the equations; but when determined from any five of a larger number of equations, then the values obtained from the five selected equations will not satisfy the others from among which they were chosen.

It is evident, however, that the larger the number of observations to determine a certain quantity, the more accurate its resulting value; so that from 32 equations based on careful observations we should get better values of A, B, C, D, E than from five.

As it is thus desirable to use a large number of observations, they must be adjusted so that the most probable

values of the coefficients shall be obtained; and this is done by the Method of Least Squares, now to be briefly explained -chiefly according to the mode of treatment by Prof. Mansfield Merriman.

305. The Principles of Probability. In mathematics, the certainty of an event is denoted by unity-its impossibility by zero; and as all its phases take place between these extremes, the probability of any one or several of these phases occurring is represented by a fraction: the numerator denotes the number of ways in which the event may happen or fail, and the denominator the total number of possible ways.

Thus, in tossing a coin, there are only two possible ways in which it can turn up-either head or tail; and the one is as likely to occur as the other: the probability of throwing a head is therefore, and of a tail, ; their sum is unity, or it is certain that the coin will turn up head or tail.

If a bag contain three red balls and five blue ones, and another bag contain four red balls and seven blue ones, the probability of drawing a red ball from the first bag is and from the second. Each bag represents the vicissitudes of certain independent events—the drawing of balls of different colors from them; but now consider them jointly—to draw a ball with each hand at the same time out of both bags: it is evident that a certain one ball of the first bag may be drawn out with each ball of the second bag and thus form a different pair every time; the same is true of a certain other ball of the first bag and all of the second; and also of a certain third ball of the first bag and all of the second; and so on until all of the first have been successively united to all of the second: this is equivalent to taking the product of the number of balls in both bags, that is 8X1188, which is the total number of different possible ways of drawing out two balls from both bags.

Reasoning in the same way, since there are three red balls in the first bag and four red ones in the second, the

probability of drawing two red ones from both bags at the same time is X; for, the total number of different possible combinations of all the balls in both bags being 88, as shown above, one of the three red balls of the first bag may be successively united to each of the four red ones in the second bag and form a different pair; then the second red ball of the first bag with each of the four red ones of the second bag; and finally, the third red ball of the first bag with each of the red ones of the second bag, which is equivalent to multiplying together their number in both bags,

and

probabilities of the primary events being separately

for the respective bags, the probability of the compound event for both bags is the product of these individual probabilities.

To consider generally a compound event made up of any number n of simple events, let p be the probability of an event happening in one trial and ƒ its failing, so that p+f=1: from what has just been stated, the probability that all will happen is p", since that of each is p, and n their number; the probability that (n-1) events will happen and one fail, in n ways, is n. p(n-1).f; similarly, for (n-2) events happening and two failing, we have, according to the algebraic theory of combination,

n(n − 1)

2

then, (p+f)" be expanded by the binomial formula, it be

+

·p(n-3).f3+etc. (150)

In this, the first term is the probability that all will happen; the second term, that (n-1) will happen and one fail; the third term, that (n− 2) will happen and two fail; and so on. If p=f=1, it corresponds to the case of throwing n coins-say 6; substituting these quantities in (150), it gives the following series as the numerical values of the several terms:

That is, if six coins be thrown, the probability of the relative numbers of heads and tails appearing in a single throw is:

That all will be heads.

That five will be heads and one tail..
That four will be heads and two tails. . ..
That three will be heads and three tails..
That two will be heads and four tails..
That one will be head and five tails.
That all will be tails...

6

And the sum of these seven probabilities is, of course, unity. If at O, Fig. 517, we erect a perpendicular Op equal to 20 parts of a scale divided into 64ths, and then on each side of it, at equal distances (e,, e2, e3), erect other perpendiculars of 15, 6, and 1 part respectively in length, and draw a curve through their ends, we shall have a graphic illustration of the probabilities given above numerically: it is called the probability curve, and is typical of all matters whose proba bilities are calculated: it is not of identical contour for all cases, but varies suitably for each, while preserving the characteristic form shown in Fig. 517.

The MOST PROBABLE event among several is that which has the greatest mathematical probability; in the above case, it is, or that in throwing six coins they will turn up equally heads and tails.