compared with like results obtained by other observers will afford an indication of the relative value of his observations.

326. From the known weights due to two independent observations there may be obtained the weight due to a phenomenon depending on those which were observed; and the following example will serve to illustrate the process. Let a certain angle c depend on two other angles A and B, and let these last be actually observed while c is obtained merely by adding A and B together, or by subtracting one of them from the other as the case may be: it is required to find the weight due to C.

E 2

1

In a single observation the precision is inversely proportional to the error: and since the weight is represented by 1 the square of the precision (art. 325.), w α or E2 α-. But, in any result obtained, like the angle c, from two observations whose errors may be represented by E, and E, and weights by w1, we, the precision varies with

1

√(E2 1 + E2 2)'

; substituting, therefore, in the

for E12 and E2, we get

In combining together several observations of circumpolar stars for determining the latitude of a station, let w, be the weight due to all the zenith distances observed when the star is above the pole, and w, the weight due to all those which are observed when it is below: then the fraction just formed will be the weight due to the result of the whole series of observations.

327. Either of the formulæ (A), (B) may be used to find the relative merits of two or more instruments: thus, let several zenith distances of a celestial body be observed with two different circles, and, omitting the degrees and minutes, let the seconds read on the head of a micrometer screw be as follow :

No. 1. circle, 58".5, 56".33, 59".25, 54".75; mean =57".2075. No. 2. circle, 40".25, 38".5, 36".8,42".5 ; mean = 39".2625. The following errors are found by taking the differences between the mean and each observation,

No. 1.,+1".2925,-0".8775, +2".0425,-2".4575. No. 2., +0.9875,-0".7625,-2".4625, + 3".2375. Taking the sum of the squares of the errors and dividing by

2.9629, and in No. 2., 4.5843, the square roots of which are 1.721 and 2.141: these divided by n (2) give 0.8605 and 1.0705, which multiplied by 2 (0.67449) give finally the numbers 0.5804 and 0.7221.

The degree of precision in the result obtained by taking the average of the first set of observations is greater than by taking that of the second in the ratio of 7221 to 5804; and the first instrument, including the skill and carefulness of the observers in both, is therefore better than the second in the ratio of those numbers. It is evident, however, that a much greater number of observations ought to be employed when considerable accuracy in the relative values is required. In like manner may the relative merits of two or more chronometers be determined, the daily rate of each for several days being considered as errors, or as observations, according as the first or second formula is employed.

The results of observations made by direct view and by reflexion with the same instrument, are frequently found to differ from one another in goodness when tried by means similar to that which has been described; and it is stated in the Introduction to the "Greenwich Observations" that, with the mural circle in use at the Observatory, the zenith distances of celestial bodies appear to be greater by reflexion-observations than by the others, the difference in some cases amounting to four seconds; near the horizon it is small or vanishes, and it is greatest when the zenith distance is about 30 or 40 degrees.

In order to find the corrections for such discordances with small trouble, the following process is used: From any point in a straight line are set out, as abscissæ, several zenith distances or polar distances, and at right angles to the line, at the termination of each distance, is drawn an ordinate equal to the error or difference between the distances obtained from the direct and reflexion observations. A curve line being traced through the extremities of the ordinates so drawn; the ordinate corresponding to any given zenith or polar distance being measured by the scale gives the value of the required error or difference, half of which is applied as a correction to the direct, and half to the reflexion observations.

328. The value of every element in physical science is determined from the results of numerous observations or experiments combined together, and, in the present state of astronomy in particular, it has become indispensable to employ some method of making the combination so as to afford the

S

most probable value of the required element. For this purpose there are formed, from the different observations, equations in each of which the true or most probable value of an element is made equal to that value which is deduced immediately from the observations, together with the corrections due to the several causes of error, those corrections involving, as unknown quantities, the most probable values of other elements.

be expressions formed from so many independent observations in which T, T2, &c. are the immediate results of the observations, a,, a,, &c., b,, b,, &c., are quantities obtained from the nature of the elements, and x and y are unknown quantities, of which the most probable values are to be determined: the terms are either positive or negative, according to circumstances. If T be the true value of the element represented by T,, T,, &c., and if all the results of observation were free from errors, we should have

T1+a1x+b1y=T, or T1-T+α, x+b1 y=0

1

2

1

1

1

T2+ aq x + b 2 y=T, or T2-T+a2 x + b2 y=0, &c. which might be put in the forms

P1+α1x+b1y=0,

Pq + a q x + b2 y=0, &c.

The nature of the terms which constitute such equations of condition will be subsequently shown (art. 450.).

It is evident that if no errors existed it would be sufficient to have only as many equations as there are unknown quantities; but, since errors do exist, the degree of correctness in the results will be greater as the number of equations is increased; and it is now to be shown how any number of such equations may be most advantageously combined.

329. A method at present much in use among astronomers consists, after having, by the necessary transpositions, rendered the coefficients of one of the two unknown quantities, as x, positive in all the equations, in arranging the equations in two groups, one of them containing all those in which the coefficients of x are the highest numbers, and the other all those in which the coefficients of x are the lowest numbers. The equations in each group are then added together, and those which are formed of the two sums are divided by the coefficients of y in each respectively; thus y has unity for a coefficient in both equations: then, if the signs of that quantity be alike, on subtracting that in which the coefficient of x is

X

the least from the other, y will be eliminated, and from the resulting equation the value of x may be found: if the signs of y be unlike, the two equations must be added together. The coefficient of x in the final equation being the greatest possible, the value of that quantity becomes the least that is consistent with the conditions, and therefore contains the smallest amount of error. If x had been determined from an ordinary mean of all the equations, its coefficients being positive in some equations and negative in others, they would, in part, have compensated each other, and thus have rendered the final coefficient less than it becomes when the process above described is employed. The value of x, found in the manner above mentioned, being substituted in the equations for y, and a mean of the resulting values of y taken; the determination of this quantity will also, since it is deduced from the most correct value of x, be obtained with the least possible amount of error.

X

This method is sometimes modified by transposing the quantities so as to render the coefficients of y positive in all the equations, and afterwards selecting from the whole those in which x has the highest positive and the highest negative coefficients of these selected equations two groups are formed, one containing the positive coefficients of x and the other the negative coefficients of the same quantity. Then taking the sums of the two groups, and dividing each sum by the coefficients of y in it; on subtracting one result from the other, y will be eliminated, and the coefficient of x in this final equation being comparatively great, the value of x will be more free from error than if it had been obtained from an ordinary mean of all. The value of y may then be found as before.

Xx

If the equations be of the form P+ax+by+cz = 0 (containing three unknown quantities) they may be divided into two sets, in one of which all the coefficients of one of the unknown quantities, as x, are positive, and in the other all are negative, and of each of these sets there may be formed two groups, one of them containing those equations in which y, for example, has the greatest coefficients, and the other those in which it has the least; thus there will be formed four groups: then, having added together the equations in each, and divided each sum by the coefficient of z in it, there will be four equations of the forms

Subtracting the third of these from the first, and the fourth from the second, z will be eliminated, and there will arise two equations of the forms

Q1 + (P1+P3) x + t1y=0

Q2+(P2+P4) x + t2 y = 0

from which, y being eliminated, x may be found in the usual way; and the coefficient of a being comparatively great, the value of that quantity will be determined with considerable accuracy. Let this value be substituted in the four equations above, and the resulting equations be formed into two pairs, in one of which the coefficients of y are the greatest, and in the other the least: then, subtracting the sum of the latter pair from that of the former, z will be eliminated; and, in the resulting equation, y having a coefficient which is comparatively great, it will be determined with considerable accuracy. With these values of x and y, that of z, found in the usual way, will also be advantageously determined.

330. A different, and, in some cases, a more accurate method of determining the most probable values of the unknown quantities, is that of least squares as it is called, which may be thus explained.

In consequence of errors presumed to exist in the observations, let E1, E2, &c. be put in place of zero in the second members of the equations of condition

E1, E2, &c. representing errors. Then imagining each equation to be squared and all to be added together, the sum of all the first members would be the sum of the squares of the errors, and there would be obtained an equation of the form

2

(P1+ a1x+b1y+c1z)2 + (Р 1⁄2 + α 1⁄2 x+by+c¿z)2+&c.=Σ.e2. Since the degree of precision in the result obtained from a given number of observations is inversely proportional to ✓.E2 (art. 324.), it follows that the precision will be the greatest when the sum of the squares of the errors is a minimum; and in order to find the values of x, y, and z, consistently with this condition, the differentials of the first member taken relatively to x, y, and z must, by the theory of maxima and minima, be made separately equal to zero. Now, on so differentiating we should get

(P1+a1x+b1y+c1z)α1+(P2+a2x+by+c2z)α2+&c.=0

1

1

1

(P1+ a1x+b1y+c1z) b1+(P2+ax+by+cqz)b2+&c.=0 (P1+a1x+b1y+c12)c1 + (P ̧ +α x+b2y+cqz)c2+&c.=0,

1