82 Choice in the Distribution of Observations 
^Vl + a(VTT^-Vr^) .N at 1, 
and hN at - (1 - VT-^'). 
a 
IX. Adjustment with regard to both of two variates connected by 
a linear relation. 
(1) The case often occurs when both of the variates observed have errors of 
observations of the same order so that adjustment only of one of them is unsatis- 
factory. We shall therefore in this section consider adjustment with regard to 
both of the variates and give the adjusted relation between them and the standard 
deviations of the constants. 
Let x' be observed with the standard deviation V aa and //' with the standard 
tleviation Vyo, we shall then for the sake of greater perspicuity exchange the 
variates for x = —~ and ti = so that both of our variates have the same 
standard deviation a. Let ^ S taken over the N pairs of observations be 
denoted by /i,, we then find, by adjusting only the y/'s according to (3), 
I X 
f^n A^iu /^2o 
!/-/^oi = ^"~^"f" (^--/^lo) (113). 
By adjusting only the x's we get 
which only coincide with (113) when 
that is when there is perfect correlation between x and // and no casual errors of 
observation. 
(2) Adjusting at the same time with regard to x and y may be transformed to 
the problem of finding the straight line for which the sum of the squared distances 
of the observed points {x, y) is a minimum. 
Let the line sought be 
X cos V + y sin (; + p = Q. 
The sum which we want to make a minimum is then 
*S' = /X20 cos^ V + jx^y, sin^ V + 2jjiT^^ cos v sin v + 2^j/X;^„ cos v + 27J^oi sin v + p-, 
dS ^ . " 
^ = 0 requu'es P ^ ~ H-10 cos v — /Xg^ sm v, 
indicating that the line passes through the mean {[Xif^, /^oi) > ^^^^ determines a 
minimum for constant v. 
