392 Dr. Wicksell : Application of Solid Hypergeometrical 



In order to derive the general relations between the 

 moments that are a consequence o£ the linearity of regression 

 we proceed thus : Multiplying (5) by 



we rind, summing for all values of f, 



i 



P21= 1 P—P3Q Or P21P 2 0=P30PU. • • • l<>) 



This formula has been deduced by Pearson in his well- 

 known memoir on the skew regression. 

 Multiplying further by 



and summing, we obtain 



PnP20=PioPm (7) 



and similarly proceeding for higher powers of f and having 

 recourse also to the equation of the other regression line, we 

 have as conditions of linear regression, 



Pa,lP20=Pa+l,0Pll, 



Pi tf iPia=Po,p+iPu ( 8 ) 



Dr. Isserlis has not deduced the moments p Bl and p u , so 

 we are not in a position to test his formula? on linearity 

 of regression otherwise than in case of the moments of the 

 third order. In case of these moments Dr. Isserlis has found 

 the identical relations 



P2l2>20l>03=Pl2p02l J Z0> 

 P02 P20 P21 Pl2 =P11 2 P0SPS0- 



Dividing these relations, we find 



P2i 2 P2i) 2z =P30 2 pn, 



Pi2 2 Pm=Po% 2 P\i- 



Taking regard of the fact that according to the results of 

 Dr. Isserlis, the moments p 2 o an d Pn as well as the moments 

 p> 50 and p 2 i have inverse signs, we see that the identities 

 contain in them the conditions of linear regression 



P21P20— Pw'pni 



Pl2P02 — P02Pll- 



