THEORY OF ERRORS. XCl 



4, 7/0, .. . SO close that we may neglect the squares, products, and higher powers 

 of A^', A7/, . . . , Taylor's series gives 



/(4 vo, U . . •) + sj^^ + % ^v + %^^ + 



— AT — A.r = o, 



which is linear with respect to the corrections Ai, Av/, .... For brevity, and for 

 the sake of conformity with notation generally used, put 



n = X — f (^o> ^70, f • • • ); 

 V = Ax, 



_^/ ;_^ r-^l 



X = Ai, y = Ar], z = AC, ■ ■ ■ ' 



Then the conditional equations will assume the form 



ax -\- dj -\- cz -\- . . . — 7i=zv\ 

 and if they are m in number they may be written individually thus : — 



a^x -\- b^y -{- c^z + . . . — Jh = ^^i> 

 - -r - -r - 1 (a) 



(fm + b„, + <^m + • • • — '^m = ^m' 



The minimum equation (5) becomes 



u = [pv"-] = [p(ax -^ by -\- cz -{-... - ^if] ; 



so that placing ^, -^, -^, . . • separately equal to zero will give as many 



dx 3y dz 

 independent equations as there are values of x, y, 2, . , . . The resulting equa- 

 tions are in the usual (Gaussian) notation of least squares : — 



[paa]x -{- [pab]y + [Z^"^] 2 + • • • — [pan] = o, 



[M] + iPbb] + Vpbc] + . . . - lpb7{\ = o, (b) 



Ipac] + Ipbc] + ipcc] + . . . - [pai-\ = o, 



The equations (a) are sometimes called observation-equations. The absolute 

 term 11 is called the observed quantity. It is always equal to the observed quan- 

 tity 7ni7itis the computed quantity/ (^0, ^0, ^ • • •)' which latter is assumed to be 

 free from errors of observation. The term v is called the residual. It is some- 

 times, though quite erroneously, replaced by zero in the equations (a). 



The equations (b) are called normal equations. They are usually formed 

 directly from equations (a) by the following process : Multiply each equation by 

 the coefficient of x and by the weight/ of the v in the same equation, and add 

 the products. The result is the first equation of (b), or the normal equation in x. 

 The normal equations in y, z, . . . are found in a similar manner. 



