TABLE 29. 

 LEAST SQUARES. 



Observation equations : 



a i z i + biz<j + . II.ZQ = MI, weight 

 . . . I 2 z q = M 2 . weight 



a n zi + bnZg + InZq = M n , weight p n . 



Auxiliary equations : 



[paa] = pia ? +p 2 ai + . . . Pna 2. 

 [pab] = piaify + p 2 a2b 2 + . . . p n a n b n . 



[paM] = piaiMi + p 2 a 2 M 2 + . . . p n a n M n . 



Normal equations : 



fpaaUi + rpablz, + . . . [pal]z q = [paM] 

 [pabjzi + [pbbjza + . . . [pbl]z q = [pbM] 



[pla]zi + [plbjzg +'...' [plljz q = [plM]. 



Solution of normal equations in the form, 



Zl = A^paM] + Bi[pbM] + . . . L^plM] 

 z 2 = A 2 [paM] + B 2 [pbM] + . . . LgtplMJ 



z q =A n [paM]+Bn[pbMJ+ ,'. . L n [plM], 

 gives : 



weight of zi = p zl = (Ai)- 1 ; probable error of zi = ^~ 



weight of za = pz 2 = (B-j)- 1 ; probable error of z 2 = - 



weight of zq = p Z(] = (Ln)" 1 ; probable error of z q = _ 

 wherein 



VR. 



r = probable error of observation of weight unity 

 = 0.6745 -%/ HX_ . (q unknowns.) 



Arithmetical mean, n observations: _ 



r = 0.6745 \j = 53 (approx.) =probable error of pb- 



\/n(n i)' servation of weight unity. 



, / Sv 2 _ 0.8453 S v / 



r = o.6745-i/ j^n: (approx.) = probable error 



\ n (n i ) nVn i of mean. 



Weighted mean, n observations: 



/Spv 2 r / Spv 2 



r = 0.6745 "Y ~n~T ; r = ^7^ =a674 5 \(n-i)Sp 



Probable error (R) of a function (Z) of several observed quantities zi, z a , . . . whose 

 probable errors are respectively, r lf r 2 , . . . . 



Z = f ( Zl , z 2 , . . .) 



Examples : 



Z = zi z 2 + . . . R2 = r\ + r\ + . 



Z = A Zl Az 2 . . . R 2 =A2 r 



Z = zi z 2 . R 2 ^zi r* + za rj 



SMITHSONIAN TABLES. 



