ALGEBRAIC SOLUTION OF normal eqUATION. 631

Log. of correlate d

added to log. in line 19,

column (d), gives the log.

of product d multiplier by (d) log.

= 7.60206 sign +

=

6.87332 sign+ No. +.001

Line 10, d multiplier by (a)log. = 6.79361 sign + No. = +.001

Line 3, d multiplier by d log.

ber =

=

7.16987 sign, Num.002. Each number is written in parenthesis under

the log. of its multiplicand.

Any line of multipliers corresponds to an equation. Take, for example, line 19: the numbers corresponding to each log. taken with the letter of the column give

The product of +0.187 (log. 9.27126) by the value of d (+.004)=+.001 (nearest unit in third place of decimals);

hence

- c + .001 c = .088.

.0890; and

Take the log. of c and proceed as with log. of d, obtaining the numbers + 0.044, line 12, and 0.029, line 5, for products of c correlate by c column multipliers.

Add algebraically the numbers on line 12 and we get the value of b = 0.486. Find its product with b multiplier line Add the numbers on line 5 for the

3, and write it on line 5. value of a.

The values of d, c, b, a can also be found from lines 28, 17, 8, 1.

For example, +4.00c -0.747d +0.3550 (line 17): substituting the value of d (+.004) and combining gives

+4.00c +0.352 = 0; and

c = 0.088, as before.

The same operation as is illustrated on page 630 can be more simply and quickly performed by the method of solution by reciprocals and Crelle's tables, instead of by use of logarithms, where the former are available; see Appendix 8, pages 26 to 28, U. S. Coast and Geodetic Survey Report for 1878.

282. Substitution in Normal Equations.-The values found for the correlates a, b, etc., must now be substituted in the normal equations (Art. 280) to test the accuracy of the solution.

For equation d (p. 629), commencing at the left:

way substitute in a, b, and c equations. In equation a, as solved above, there is an error amounting to .001, but as only corrections to the nearest hundredth are desired, this small residual may be neglected.

283. Substitution in Table of Correlates. The values of the correlatives are next placed at the head of columns A, B, C, and D (p. 628, Table of Correlates Solved), and products by corresponding coefficient in column a, b, c, or d, of the adjoining table are found and written on the proper lines in A, B, C, and D columns.

The sums of the products on each horizontal line are placed in the column of totals. As a check on this part

of the work see that the sum of the numbers in each column = 0.

On each line in the column of totals is the correction for

WEIGHTED obserVATIONS.

633

the side adjacent to an angle, the numbers for which are given in the column of sides.

For the angle 1.2.3. = (1.23.2) the correction is as

follows:

For the side 1.2 (line 4) the correction is - 1.447.

For the side 3.2 (line 5) the correction is + .0962.
Hence for the sides

1.2 and 3.2 we have

These are the corrections which are written in the third and fourth columns of the angle equation (Art. 276, p. 620) as side corrections and angle corrections respectively. From their application result the corrected spherical angles, column five.

The correction for each log. sine is the product of its angle correction by its difference for 1". For example, sine of angle 3.4.1 (Art. 279, p. 626), Dif. for 1" = 5.9, column three, correction for angle = -o".66, column four.

5.9 X 0.66 +3.9 cor. to sine, column five.

284. Weighted Observations. Where a number of observations differ somewhat from each other and the causes are believed to be known for such difference, it occasionally becomes desirable to give greater value to one observation than to another; thus one may be given two or three times the value of another. This operation is called weighting (Art. 264). To find the weighted mean of a number of observations which have been given unequal weights, each is multiplied by its proper weight, and the sum of the product is divided by the sum of the weights, the quotient being the weighted mean.

Weighted mean = 38° 54′ 55′′.4

Weights are used in a least-square adjustment in the following manner: the adjustment is carried forward as above described till the table of correlates is reached; then opposite each angle number in a station adjustment, or opposite the side numbers in a figure adjustment, place the weight of the angle or side in a separate column. Every product formed in the table of correlates must be divided by the weight written on the horizontal line with the multiplicand. The weight is used only as a divisor.

The following example from a station adjustment will illustrate the method of using weights in station or figure adjustment.

Equation a, at the bottom of page 635, is formed from the left half of the table on the same page, thus:

Term b, same equation is from second line.