TABLE 3O. 

 LEAST SQUARES. 



Observation equations : 



aizi + b!Z 2 + . . . liz q = M b weight p t 

 a 2 zi + b 2 z 2 + . . . I 2 z q = M 2 . weight p 2 



a n z! + b n z 2 + . . . l n zq = M n , weight p n . 



Auxiliary equations : 



[paa] = piaf + P2af + p n a^. 

 [pab] = Piaib! + p 2 a 2 b 2 + . . . p n a n b n . 



[paM] = piaiMi -f p 2 a 2 M 2 -f . . . p n a n M n . 



Normal equations : 



fpaa]zi-f [pab]z 2 + . . . [pal]z q = [paM] 

 pabJ Zl + [pbb]z2 + . . . [pbl]z q = [pbM] 



[pla] Zl + [plb]z 2 + . . .' [plljzq = 



Solution of normal equations in the form, 



Zl = AJpaM] + BifpbM] -f . . . I 

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



zq = A n [paM] + B n [pbMJ + .'. . L n [plM], 

 gives : 



weight of zi = pzi = (Aj) 1 ; probable error of zi = - 



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



\/Pz 2 



weight of z q = p Zq = (Ln)" 1 ; probable error of z q = 



wherein 



r = probable error of observation of weight unity 



= 0.6745 -i/ (q unknowns.) 



Arithmetical mean, n observations: _ 



r = 0.6745 A/ (approx.) =probable error of ob- 



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



/ S V2 _ 0.8453 2 V 



-- 



V _ 0.453 V 



r = 0.67 45\/ : -- 7==. (approx.) = probable error 

 \n(n-i) n Vn-i of mean. 



Weighted mean, n observations: 



/Spv2 r 



r = 0.6745 \ ~ ; r = -==o. 



Probable error (R) of a function (Z) of several observed quantities zi, z 2 , . . . whose 



probable errors are respectively, i - i, r.> 



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



Examples : 



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



Z = Az! Bza . . . R 2 =A 2 r\ + 



Z = zi z 2 . R- = z i 2 r!5-fj 



SMITHSONIAN TABLES. 



