THEORY OF ERRORS. XCUl 



£„, = \ l-P^,. for unequal weights, 



\ M — fX 



■— ./ IT'^J for equal weights, 

 V w — /.I 



Mean error of any observed quantity (n) of weight/ = -i_' 

 Mean error of .v = e,„ y/uj, 

 Mean error of j = e„, ^yf^j, 

 Mean error of s = e,„ y'yy, 



where aj, /?,., 73, ... are defined by equations (c) and (d) above. 



e. Case of functions of several observed quantities x, y, z, . . . . 



This case is that in which the conditional equations (4) contain no disposable 

 quantities $, rj, C, . . . . It is the opposite extreme to that represented by the case 

 of the preceding section.* It finds its most important and extensive application 

 in the adjustment of triangulation, wherein the observed quantities are the angles 

 and bases of the triangulation, and the conditions (4) arise from the geometrical 

 relations which the observed quantities />///s their respective corrections must 

 satisfy. 



An outline of the general method of procedure in this case is the following : — 

 The first step consists in stating the conditional equations and in reducing 

 them to the linear form if they are not originally so. The form in which they 

 present themselves is (4) with $, -q, (, . . . suppressed, or 



wherein x, y, z, . . . of (4) are replaced by x^, x^, x^ ... for the purpose of sim- 

 plicity in the sequel. If this equation is not linear, Taylor's series gives 



SF 9F 



F(xi, xo, Xs. . .)-\- g^ Axi + -g^^ A;c2 = . . . = o, 



since the method supposes that the squares, products, etc., of A^Cj, Ax., . . . may 

 be neglected. The last equation is then linear with respect to the corrections 

 Axj, A^iTj . • . which it is desired to find. 



For brevity put 



F(xi, X2, 0.3 . . . ) = (7i, a known quantity. 



OF 9F 9F 



»^3 



9x, — ''" 9x, — "'-^ 9x^ — ""■'' 



Then the conditional equations will be of the type 



ai^Xi -f- '5',.-^-V2 -|- a-Axs + • • • + !7i = o. 



* The middle ground between these extremes has been little explored ; indeed, most practical 

 applications fall at one or the other of the extremes. 



