KiRSTiNE Smith 
83 
The corresponding S is 
= - ^?u) cos2 V + (/X02 - /x5i) sin2 v + 2 (/x,i - ^u-m/Xio) cos v sin v (115), 
which differentiated with regard to v gives 
= - {fX2o - /^ui - (h^o-i - ^oi)} sin 2v + 2 - /Xoi/Xm) cos 2v. 
It thus follows that 
2 - /^oij"io) _ 2 tan u 
or 
tan V = - 
tan 2w = \ - 1 J. 2 ' 
/^2o - f^lo - (/^02 - Mm) 1 - tan^ V 
^ {/^02 - /^m - (/^20 - /^l'(i) ±^[^^02 -/^oi - (/^20 - /^lo)]^ + Hh-u - l^wl^oif} 
1^11 /^Ol/^IO 
(116) 
determine a maximum and a minimum of S. 
Substituting in (115) we find 
{(^20 - MIo + l"02 - f^m ± V'f/Xoo - fAt> - (H-02 - l^l\W + 4 [fill - /^oiMio]^} > 
so that the minimum corresponds to the negative sign of the root in (116). 
The adjusted function connecting x and y is hence a line through the general 
mean forming an angle u with the x-axis which is determined by 
tan M = - cot V = - i^m - (M20 - + V^[/^20 - Mio - (/^02 - l^m)f + 4 l>ii - 
(117). 
For the variates x' and there must to this value of the tangent be added the 
factor expressed by the moment coefficients of x' and y' we therefore find 
«(/^..;i-f<'5i)-y(/i2»-/ii5) + '^/[y(M20-Mid) -a(Mw-/ioi)] + 4ay [^i,-^„i/xi„]2 
tan u — — — ; — J- 
2a (/An - /Aoiftio) 
(118). 
(3) We shall prove that the line is situated between the two regression curves 
(113) and (114). 
Making {^x^q, jLtgi) the zero point of the coordinates, the three tangents to be com- 
pared are 
^-^^ and ^ {/X02 - /i2o + V(/Xo2 - M2o)2 + = tan M, 
P20 Mil -'Ml] 
where the jxs now are the moment coefficients about the mean. 
According to ^j.^^ 5 0 we have 
Mil <^ M02 
M20 Mil 
since /x;, < /xgo . Mo2- 
As V (^02 - M2o)^ + < /Ao2 + M20 3 
we have tan m $ ^ . 
Mil 
