THEORY OF ERRORS. Ixxxvii 



equations. Thus, let .v, y, c, ... he the observed quantities with weights /, ^, 

 /-.... Let the corrections to the observed quantities be denoted by A^v, Aj, 

 Ai;, . . . ; so that the corrected quantities are x -|- Ax, _>/ -j- A^, 2 -}- A0, . . . . Let 

 the disposable quantities whose values are to be determined along with the correc- 

 tions be denoted by ^^ 7;, ^, Then, the theoretical conditions which must be 



satisfied by x + ^x,'}> + Aj', s -\- Az, . . . and by i, vA, - ■ • may be symbolized 



^ J^u a, v,^,'-- -V + ^^,y-\- ^y, - + Az,...) = o. (4) 



Subject to the conditions specified by the // equations (4), we must also have 



/ (Axy + ^ (Ajf + /- (As)'" + • • • = a minimum (5) 



= u, say. 



Equations (4) and (5) contain the solution of every problem of adjustment by 

 the method of least squares. Two examples may suffice to illustrate their use. 



First, take the case of the observed angles of a triangle alluded to above. 

 Calling the observed angles x, y, z, we have 



x-\-Ax-\-y-\-Ay-\-z-\-Az=:^ 180° + spherical excess, 



or 



Ajc -\- Aj -|- As: = 180° + spherical excess — (pc -\- y -\- z) 



= c, say. 



This is the only condition of the form (4). The problem is completely stated, 

 then, in the two equations 



Ax -\- Ay -\- Az = c 

 , p {Ax)" + q {AyJ- -f r {Azf — a min. = ti. 



To solve this problem the simplest mode of procedure is to eliminate one of the 

 corrections by means of the first equation and then make u a minimum. Thus, 

 eliminating As, there results 



u=p (Axy + q (Ayf ^r(c- Ax- Ay)\ 

 The conditions for a minimum of it are : — 



=. (J> -\- r) Ax -\- rAy — re = o, 



-^^ = rAx -\- {q -\- r) Ay — re = o ; 



and these give, in connection with the value Az :^ e — Ax — A_y, 



Ax =2, Ay=Q, Az=Q. 



where 



- + - + - 



p^ q^ r 



When the weights are equal, or when p ^ q =z r, the corrections are — 



Ax = A^ = As = ^ c. 



5ax 



dil 



