Table 29. 

 LEAST SQUARES. 



59 



Observation equations : 



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

 a2Zi + b2Z2 + . . . l2Zq = Mj. weight p2 



Auxiliary equations : 



Normal equations : 



anZi + biiZo + . . . InZq = Mn, weight pn. 



[paa] =piar +P2a| + . . • pna^. 

 [pab] = piaibi + P2a2b2 + . . . pna„bn. 



[paM] = piaiMi + paagMa + . . . pnanMn. 



[paa]zi+ [pab]z2 + . . . [pal]zq = [paM] 

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



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



Solution of normal equations in the form, 



zi = Ai[paM] + Bi[pbM] + • • • Li[plM] 

 Z2 = A2[paM] + B2[pbM] + . . . L2[plM] 



zq = An[paM] + Bn[pbMi + •"• ■ Ln[plM]. 



gives 



wherein 



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



VPZl 



r 



weight of Z2 = pz2 = (62)"^ ; probable error of Z2 = 



V/Pig 



weight of Zq = pz = (Ln)~^; probable error of Zq = — — 



VPz^ 



r = probable error of observation of weight unity 

 = 0.6745 -%/ — !— . (q unknowns.) 



-q 



Arithmetical mean, n observations : 



= 0.6745 a/ — — = - ' (approx.) =probable error of ob- 



' "~^ V^n(n— i)' servation of weight unity. 



, / S v2 _ 0.845 3 2 V , 



ro = 0.6745-1/ — : ' /i • (approx.) = probable error 



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



Weighted mean, n observations: 



, /Spv2 r ^ / Spv2 



r = 0.6745 V ^^: ^o =^^=0.6745 ^^^ 



i)Sp 



Probable error (R) of a function (Z) of several observed quantities zi, Z2, . . . whose 

 probable errors are respectively, r^, r2, . . . . 



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



Examples : 



Z = zi ± Z2 + . . . Ri = rl + tI+ . . . 



Z = Azi ± Az2 ± . . . R2 =A2 rl + Bhl+ . ."". 



Z = zi Z2. R* =zi r^ + Zi r\. 



Smithsonian Tables. 



