KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58. N:0 3. 29 



The result will be found satisfactory. 



If we vvrite the indices backwards and interchange r and i" the fornuilae (40) 

 will give also the value of o}(r) and thus the curve of scedasticity of rj on §. 



In the following the formulae of the regression of the moments of the third 

 and fourth order in the arrays will be deduced. On account of the complexity of 

 the algebraic developments only ease I will be regarded. It will be found that in 

 the same approximation as before the characteristics of the third order will have 

 linear regression, and the characteristics of the fourth order will have no regression 

 at all but have a constant value in all the arrays. The same result will hold for 

 case II. It will also be supposed, when nothing is stated to the contrary, that 1— r 2 

 is a quantity that is not very small (of the order of magnitude of the /?</). 



(9) The regression of the characteristics of the third order. 



Here we have 



''' 3 (?)',= fi*F(lri)dS:fF(§,ri)di, 



and further 



(42) M!), = ^,(!),-3<r$(!).i„-ä5. 

 We find 



:i[ - hl 7<°> + 8 7 (0) 4- />', / (0) + • ■ ■ 



This leads to the formula 



v' 1 (|), = [-3fÄ 1 (i ? )-f»Ä,(ry)-A,(3rÄ 1 (i ? ) I r»l^))-/* 04 (3rÄ s ft) + r»Ä 7 ft))- 



(43) Zt*yARAv) + r*BAv))-3P l ARAri) + r*BAy))-Grp n RAv)-Gr(i„R 3 (v)- 

 -6/?,, - 6/*„ R, (,)] : [1 + p„B,(n) + ,>•„, Ä : (,,)]■ 



In case II the terms - * fi\, (3 r R 7 fo) f r 3 R a (rj)) — 3 ft, /*„ (Ä 5 (,) + r»Ä 7 (»;)) - 



— 6 ('"'o. .>'.-! +,, ^JriM'?) — 6 (fAio #>.■( + i''rjf>7i) K A.O should be added to the numerator and 



the term + />'„„ Ä c (/ ( .) to the denominator. 



Developing the denominator and arranging in powers of ', we obtain 



'''3m, = — (6ft, — 3/5f„ -6r/* 2I I- 9r/? 03 + 9r 2 ft 2 - 15r 3 /? 0S ) + 



4- ,, (3r + 6/»,, - 3r 3 — Op',, — 18r/tf 22 + 36 r t i 0l + 45r»^ 13 — 96 r 3 />'„,) - 



(44) —^(3^4 Orft, — 9f/J„-18r«/?i» + 36r«|« 0l ) + 



+ j; 3 (r 3 + 3 /? 13 + 6 r £ 22 — 12 r p\ l4 — 30 r 2 /?„ + 84 r 3 p oi ) — 

 -tfXSr*fc t -9r*fi„) + 

 + ^(3^/^3-12^^,). 



