XCii THEORY OF ERRORS. 



A noteworthy peculiarity of the normal equations is their symmetry. Hence in 

 forming equations (b) from (a) it is not essential to compute all the coefficients of 

 X, y, 2, . . . except in the first equation. 



Checks on the computed values of the numerical terms in the normal equations 

 are found thus : Add the coefficients a, b, <-,... of x, 7, s, ... in (a) and put 



^^i + ^1 + ^1 + • • • = -^b 

 a-i -\- b., -\- c, A^ . . . z:^ Jo, 



Multiply each of these, first, by \\.s pa ; secondly, by \\.s pb, etc., and then add the 

 products. The results are 



\^pad\ -\- \_pab'\ -\- [pac] -\- . . . = [pas] 

 [pab] + [pbb] + [pbc] + . . . = [pbs] 



These will check the coefficients of x, y, s, . . . in (b). To check the absolute 



terms, multiply each of the above sums by its ///, and ?dd the products. The 



result is 



[pan] 4- [pb?i] + [pm] -{- . . . = [psn], 



which must be satisfied if the absolute terms are correct. 



Checks on the computation of x, y, z, . . . from (b) and of z'j, v.^, . . . from (a) 

 are furnished by 



[pav] = o, [pb7'] = o, [pcv] = 0, .... 



To get the unknowns x, y, z, and their weights simultaneously, the best method 

 of procedure is, in general, the following : For brevity replace the absolute terms 

 in (b) by A, B, C, . . . respectively. Then the solution of (b) will be expressed 

 by 



y = a. + /?., -f- y, + . . . , (c) 



Z=a.i + /?3 +73 + . . . , 



in which oj, ^1, 71, . . . are numerical quantities ; and 



weight of .r = — > 



weight of V = 7T > (d) 



P2 



weijrht of ^ ^ — > 



To compute mean (and hence probable) errors the following formulas apply : — 

 m =z the number of observed quantities n 



= number of equations of condition, 

 fx. = number of the quantities x, y, z, . . . 

 €,„ = mean error of an observed quantity («) of weight unity, 

 €p = corresponding probable error = 0.6745 c^. 



