Table 30. 59 



LEAST SQUARES. 



Observation equations : 



aizi + biZ2 + . . . liZq =Mi, weight pi 

 a^zi + b2Z2 4- . . . hH ~ ^2- weight pa 



anZi + bnZ2 + . . . InZq = Mn, Weight pn. 



Auxiliary equations : 



[paa] = pia? +P2a| + • • • Pna^. 

 [pab] = Piaibi + P2a2b2 + • • • Pnanbn. 



[paM] = piaiMi + p2a2M2 + • • • PnanMa. 



Normal equations : 



[paajzj + [pabjzo + • • • [pal]zq = [paM] 

 [pabjzi+ [pbbjzo + . . . [pbl]zq = [pbM] 



[pla]zi + iplb]z2 + . . ." [plljzq = [plM]. 



Solution of normal equations in the form, 



Z1-- AifpaM] + Bi[pbM] + . . . Li[plM] 

 Z2 = AoLpaM] + BaLpbM] + . . . LsLplM] 



gives : 



wherein 



zq = An[paM] + Bn[pbMJ + . . . Ln[plM], 



weight of zi = pzi = (Ai)-' ; probable error of zi = — ^^ 



VPzt 

 r 



weight of Z2 = pz2 — CB^)-^ ; probable error of Z2 = — ^_ 



VPxa 



r 

 weight of zq = pz = (Ln)~'; probable error of Zq = — z:^. 



VPz„ 



r = probable error of observation of weight unity 

 = 0.6745-%/ — (q unknowns.) 



-q 



Arithmetical mean, n observations : 



r = 0.6745 \l = - (approx.) =probable error of ob- 



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



2 v2 _ 0.8453 S V , , , , , 



ro = 0.6745-1/ — ^/_ — • (approx ) = probable 



\n(^n— i; nVn— i of mean. 



Weighted mean, n observations: 



r = 0.6745 V ^^; ro =^;=^=o.6745 ^J^^Z^yTp 



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

 probable errors are respectively, rj, rj, . . . . 



Z = f (Zi, Z2, . . .) 



Smithsonian Tables. 



