KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58- NIO 3. 47 



fractional function in whicli the nnmerator and the denominator are developed in 

 polynomes of Hermite. Thus v' s (x) y being any moment of the y-arrays as taken about 

 the total mean of x, we have 



a a + a l R l (y) + a 2 R 2 (y) + a x R,(i/) + a,h\l!/) ■«,/?..(//) -t- ■ ■ ■ 

 («) * «(*)» = ~ 



(i*) 



1 c 2 Ä 3 (y) + c, Ä 4 (t/) + -- 



The characteristics are tlien proportional to the functions 



a? 3 = >'',(.»)„, 



or£(a;) = r'Jx),, - v\{x)l, 



v 3 (x) y = ^ a (a-)„ — 3 r,(r), y ^(a:),— v\(x) 3 v , 



/., {x) y = i>' 4 (x)y — 4 »/,(*)» v\{x) y — 6 >',(x)„ >-\(x)l — v\{x) y — 3 v 2 ( i 



]lt 



order 



Ond 



» 



3"' 



» 



4"' 



» 



Of course the expressions («) as well as (jff) may be written in the form of 

 fractional functions in which the numerator and denominator are arranged in powers 

 of y. Thus any characteristic is given by the ratio of two power-series in y. 

 However, it will here be natural to retain the arrangement in polynoms of Hermite, 

 by which the characteristics of higher order than the first will contain squares and 

 products of the polynoms in both numerator and denominator. 



Two different cases of approximation occur whcn moments up to the fourth 

 order are considered. They are referred to as case I and II. No difficulty whatever 

 presents itself in deciding which case we have to deal with in a special instance. 



If the denominator of («) or (,j) be expanded the same two cases must be 

 considered. It is, however, most advantageous to split up the polynoms of Hermite 

 and arrange according to powers of y. We then get for the characteristics expressions 

 of the forms 



x y — m l = b | b l (y- m 2 )H b 2 (y m .)- t h.(u -m,) s , 



Oy{x) -6' 8 i b\(y m t ) + b' 2 (y -vt r 



r„{x) :i =b Q " + b" l {y-m 2 ), 



K(*)v = J . 



ni 



The coefficients are in both cases (I and II) expressed in terms of the charac- 

 teristics a lt ä 2 , r, fy of the correlation surface. The neglected terms are in all the 

 equations of the same absolute order of magnitude, being always of a smaller order 

 than PijPpq where in case I i + j + p + q>4: and in case II i + j + ]j + g>6. In addition 

 characteristics § tj for i J rj = 5 are always neglected. 



The range of applicability of formulae (y) is theoretically smaller than, or at 

 most as great as, the corresponding range for formulae (a) and (/?). In practice, 

 however, we find that the range is generally large enough, and that at times it is 

 even greater than for formulae («) and (/?). The latter case occurs, I presume, when 

 the regressions are not far from linear, as then (we speak here only of the regression 



