90 C. E. VAN ORSTRAND 
The arithmetic mean of the gradients is 
— nfm + 2/2 + ++ + Yn/ Kn 
nN 
b (2) 
and assuming that the weight of each value of 5 is proportional to the 
depth, we have from 1, 
ab = 1 2b = Yo- > + Xnb = Yn 
which gives for the weighted mean, 
Ji 2 
Equations 2 and 3 are the ones in general use. In order to deter- 
mine the real significance of these equations, let us consider the 
adjustment of a straight line through the origin. Let it be assumed 
that we have n observation equations: 
b (3) 
1b = sn weight pi 
xeb = yo weight po 
oe al te) Peraie® “oy ‘el? & a (4) 
nb = yn Weight pp 
the solution of which by the method of least squares gives 
; Pidiyi + prreye + ++ > Pn®nVn (s) 
SSS 5 
pier? + pox? + + + + Pdr? 
the residuals (21, 2, . . . mm) being subject to the condition 
P1%i1 + porate + - + + PatnPn = 0 (6) 
Following are important special cases of 5: 
1/%1 + ya/%2 + ++ - n/n I I I 
b = ——____-——_- », = — pp = —--- pp =— (7) 
n x12 x92 Xn? 
Vit Va a at va I I I 
b See eee = — ==...) = 
Rite Mate t rs Me : a i Xe ; Xn 4 
iN + Mee + + * Kan 
b unCtl eC CO = — Cyn Wak n 
Ga at Ja gh Oe pi = prs p (9) 
x71 = 2" Vo S Sen 
’ = A= 1% pz = % °° * Pn = Xn (ro) 
Sl a le ei 
750 
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