KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58- N:0 3. 11 



If the E are small we may put instead 



8x p = yx p + %(«/)%(#), 



Plotting ö Xp and ö Uq against x p and y q as arguments we obtain the regression 

 curves of the modes. Theoretically these are the really adequate regression curves, 

 as they give the most probable value of the one variate for a fixed value of the 

 other. Practically, however, they will be of but little use, as their equations are much 

 more complicated t han the equations for the regression of the means. 



The regression curves of the modes may also be obtained from the equations 



</F(z, <u = 



6å t 



<>F(dy, y ) 



ddy 



= 0. 



In this fashion we get implicit equations for the curves, while the regression 

 of all the other characteristics is given by explicit functions. If we attempt to 

 develop the roots of the above implicit functions in powers of the argument, there 

 will be difficulties of convergency, as Chaelier has shown (in a paper as vet un- 

 published). 



From what has already been said on the convergency of the yl-series as a 

 function of interpolation for arbitrary functions, we may conclude that the distribu- 

 tions in the arrays may be given the following forms: 



* * p (?/) = <P* P (V) ^ 3 Sr p o Xp —yp- + 3 E Xp a% p —^- + ■ . 

 and 



1 d S( flJnW 1 '' i( Pu„( X ) 



Here we must take 



and 



1 g 



f P* p (y) = 77^ C °*V , 



o Xp V27t 



( x - x y q Y 



9V,(*) = ^ e ~ 2 ° u « • 



o,. V2tc 



By the aid of these equations it will be possible to assign a certain probability that 

 y will not deviate by more than a given quantity from y r when x has the value x p ; 

 and vice versa that x will not deviate by more than a given quantity from x y when y 

 has the value y q . 



