KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58. N:0 3. 33 



(11) The clitic curves, the synagic curves and the curves of regression of ihe 

 7iiodes in the arrays. 



The skewness of the distribution in the ?j-arrays is given by 



&(!) = - 



r 3 ($)„ 



'k 



2.a°(5) 



From (40 I) and (46) we obtain 



,_,, o m = o fto — rfti + *•'/?,» -r 3 fl.,-»j(ft. — 2rft, + 3rV lit - 4rV 04 ) 



( ' "^' J [l-r 2 -2^ 2 + 6r^ 3 -12r^ 04 - ) ;(2^ 1 -4r/i 13 + 6r / i 03 )+r !! (2^ 2 -6r^ 13 +12r^ 04 )] ! 



If &,(£) is plotted against », (or y) as argument the clitic curve is obtained. 

 The excess of the distribution in an »j-array is given by 



Hence we obtain from (40 I) and (53) 



1 ' Ä " fl5 ' [l - r 8 — 2& 2 ;• 6r/S n - I2r*p 0i — rj(2^ 2l - 4rfc, f 6r 8 /? 03 )+if (2/* 22 — 6r/? J3 + 12r 8 /* M )]*" 



The denominator may also (if l--» 2 is not very small) be taken from (52). 



If E v (§) is plotted against *, as argument we obtain the synagic curve. 



In deducing the formulae for the regression of the characteristics of the third 

 and fourth order we have made the approximation under the assumption that 1 — r 2 

 is a quantity of greater order of magnitude than the /?y. If the correlation is mo- 

 derately anormal this does not prevent r from being considerably great, say, upwards 

 of 0,7. In such a case it will be consistent with our previous principles of approxi- 

 mation if we write 



(56) £,(!) = - 7 =L= i¥ [/!? 30 -r/? 21 I r 8 /? 12 r*p 03 -r)((l 3l -2rp 22 + 3r^ 13 -4rVoJJ, 



V \ — r- 



(57) &,(!) = - - Tj[fi„-rp„ + r 8 /?„-r^ ls + r*/? 04 ]. 



( 1 — r') 



The distances from the means i",, to the modes of the arrays will then, in the 

 same degree of approximation, be 



(58) J v (§) = ™- (& - rp tl + r 8 /? I2 - r 3 ^ 0s - , (fo, - 2r& 2 + 3ry i3 - 4r*/? M )] . 



Hence, for the regression of the modes, or, which is the same thing, for the 

 equation connecting any fixed value of r\ with the most probable value of f asso- 

 ciated with it, we obtain the cubic 



K. Sv. Vet. Akad Handl. Band 58. N:o 3. 5 



