The adjustment of observations by the method of least squares; with applications to geodetic work

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 Excerpt: ...x + pbb y+... = pbl, (2) we should have x = aj = una pal + uafi pbl + y = # = MB0 M + x,m pbq +... (3) where x, y,... are the most probable values of the unknowns. We must, therefore, express A2 in terms of the residuals v in order to find fi. From the two sets of equations, by subtracting in pairs, = vx + ax (x0-x) + bx (y0-y) +... = v2 + a2 (xo-) + b2 (y»-y) + (3) Squaring, adding, and attending to equations (9), Art. 97, we have approximately, n( being the number of unknowns, A2 = tt + ntf. (4) Putting A2 = «/, there results the expression required. Reasoning as in Peters' formula, Art. 32, we easily deduce from (4) r =.8653-f==., (6) which is known as Liiroth's formula (Astron. Nachr., 1740). When n = I, equations (5) and (6) reduce to Bessel's and Peters' formulas respectively (Arts. 29, 32). 106. Methods of Computing V.--(a) The ordinary method is to substitute the values of the unknowns found from the solution of the normal equations in the observation equations, and thence find vv v2,... The sum of the squares of these residuals will give V. The residuals having to be found, for the purpose of testing the quality of the work this method of computing V is on the whole as short as any. As checks on the values of V found in this way the following are of value: (b) If we multiply each observation equation by its v and take the sum of the products, then, remembering that av = 0, bv =0,..., we find M =-«/. (c) If we multiply each of the observation equations by its / and take the sum of the products, alx+bly+ ff = trfl =-t. (d) We have for two unknowns, x and y, v' = (ax + by-If = aa x+ 2 ad xy + 6b f-2 al x-2 bfy + 11 V aa ««/ v M I and generally for w unknowns, ry = (o» + &y + O2 Now, from (9), Art. 85, the coefficients...