XC THEORY OF ERRORS. 



A formula for the average error sometimes useful is 



Average error = // — ^;^ n r .-i for unequal weights. 



= ^m {m - 1) f°' ^^"^^ ^'^'g^^^^- 

 In these the residuals v are all taken with the same sign. A sufficient approxi- 



[7'! 



mation in many cases of equal weights is — ; but the above formulas dependent 



on the squares of the residuals are in general more precise. 



An important check on the computation of x is [/z'] ^ o ; /. e., the sum of the 

 residuals v, each multiplied by its weight, is zero if the computation is correct. 



d. Case of observed function of several unknown quantities i, ■>]> C • • • • 



A case of frequent occurrence, and one which includes the preceding case, is 

 that in whicli a function of several unknown quantities is observed. Thus, for 

 example, the observed time of passage of a star across the middle thread of a 

 transit instrument is a function of the azimuth and collimation of the transit 

 instrument and the error of the timepiece used. In cases of this kind the con- 

 ditional equations of the type (4) assume the form 



that is, each of them contains but one observed quantity x along with several 

 disposable (disposable in satisfying the minimum equation) quantities ^, ?;, ^ . . . . 



The process of solution in this case consists in eliminating the corrections 

 Aa-,, A^2j • • ■ from the above conditional equations, substituting their values in 

 the minimum equation (5), and then placing the differential coefficients of u with 

 respect to ^, -q, ^ . . . separately equal to zero. There will thus result as many 

 independent equations as there are unknown quantities of the class in which ^, rj, 

 C ■ ■ ■ fall, the remaining unknown quantities Aa-j, A^Cg^ . • . , or the corrections to 

 the observed values, are then found from the conditional equations. 



In many applications it happens that the conditional equations 



are not of the linear form. But they may be rendered linear in the following 

 manner. First, eliminate the quantities x -\- Ax from the conditional equations. 

 The result of this elimination may be written 



/(^, V,C---)-x-Ax = o. 

 Secondly, put 



V = V'> + ^v, 



where $0, %> • • • are approximate values of ^, 7/, ... , found in any manner, and 

 A^, Ary, . . . are corrections thereto. Then supposing the approximate values 



