KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58- N:0 3- 35 



Now, this requires that the /?,-> shall vanish whenever i or j are equal to 1 or 2, and 

 further that r = 0;^ pa = ^po.^og for p and g>2. Thus we conclude, with aid of the 

 exact formula (32), that independency of the variates will make |,,= and »;i = 0. 

 But this conclusion cannot be reversed, because the /?«2 and foj, i and j>l, do not 

 enter into the exact formula of regression of the means. Thus, while £,, = 0; r^ = 

 makes it necessary that @n and fa as well as r should be zero, the values of some 

 of the fa and fa may still be considerable, and the variates then cannot be separa- 

 ted in the correlation function. 



From our formulae we conclude that: 



1. A necessary and sufficient condition for independency of the variates is 

 that r = 0; fa = for * or j equal to 1 or 2; fa = fafa for i or j>2. 



2. A necessary condition for the regression of the means to be parallel to the 

 axes, that is, for the means in the arrays to be independent of the arguments of 

 the arrays, is that 



r = 0; p l2 = 0; p 9l = 0; /? 31 = 0; /?„ = etc. 



3. A condition for linearity of regression is that 



r 30 = 0; r 40 = for the one curve, 



>'nj~ 0; r 0i = >> » other » 



4. An approximative condition for homoscedasticity is that 



,/.., - 2r,i l2 + 3r>fa = 0; fa - 3r/? 13 + 6r*fi„ - 0, 

 p ii -2rfa + 3r*fa = 0; fa — Sr fa + Hr,)',, . -= 0. 



5. An approximative condition for homoclisy is that 



/? 31 -2r/* 22 + 3r 2 /? 13 -4r s jS 04 ==0 

 p l3 — 2rfa + 3f"A,--4f»jJ 4é = 0. 



Conditions 4, 5 are only valid as far as our approximations of case I are appli- 

 cable. In 3 the conditions are valid in so far as case I or case II of approximation 

 is sufficient. Conditions 1 and 2 are not subject to any restrictions. 



(13) Exam/ples. 



Example 1. Case of skew correlation with linear regression and parabolic sce- 

 dasticity. Number of trumps on the first two hands in 25000 deals of whist. Ex- 

 periment by Pearson, recorded by L. Isserlis Phil. Mag. Sept. 1914. 



The number of trumps were distributed according to the following table. 



