448 Prof. S. D. Wicksell on the Normal Correlation 



We have now, remembering that in the case under 

 consideration r pq = rq p and r vp —l, 



H200....0 = [Xn q w q ] 2 — l, 



HllO = %ri q Wq'Zr2qlCq ?*12> 



H400....0 = [1ri q io q Y — 6[2ri q w q ~\ 2 + 'd, 



H310....O = [%riqWq] S .'2<r2qWq — 3%r\ q W q .%V2 q W q 



— 3 [%riqWq] 2 H2 + 3^12, 

 H220 .... = [triqlUq] 2 [Xr2qWq] * — [X^qWq] 2 — [Zn q W q ] '' 



— 4712 Zn q lV q . 2<r2qWq + 1 + 2?i2 2 . 



H 



2110 



[2n<zW ff ] 2 2r2 2 M> 2 .2?'3gWg — %r2qWqtr 3 qWq f' W 



= rs* 



— 2r\2 %r\qWq Xr3qWq — 2ris XriqlOq Sr2qWq 



— T23 [XriqiUq] 2 + r 2 3 + 2?-2i ri 3 , 



H HllO .... O = XriqWq . ^r 2 qWq . %>'3ql0q X^qWq 



— ri2 ^r 3q Wq 2r4 2 w\ z — ri3 2r2 2 w 2 %r^ q w q 



— T23 XnqlVq %V4 q W q -ru%r 2 qWq ^^qWq 



— ?'24 XriqWq %r3qlVq — ^ Xn q W q %r2 q lV q 



+ »'12 ?'34 + ?13 ?*24 + ^23 ^14- 



Putting 



J - 00 J -co J - cx> 



we find consequently 



,x n ) = m kJ . 



™20 . 

 "'no. 

 ^40 . 

 ^310. 



^220 . 

 W2110 

 WllllO 



.0 = l, 



.0 = n.2, 



.0 =3, 



. = 3?12, 



.0 = 1 + 2ri2 2 , 

 ...0 = r23 — 2ri 2 ri3, 



. . . = ?*12^34 + ^13^24 + ^23?'14 



(6) 



and any other moment of the second and fourth orders may 

 be obtained by permutation of indices. 



Without deducing the polynoms H of the sixth order, 

 the moments of the sixth order may be determined in the 

 following way : — 



Evidently we have 



Jen 



H /;A . . . . k s + 1 . . . , lc n = 1 (j ^'" ""' - 2 V^ 11 V< 2 • • • • **' 



