XX VII XXX] Iniroiltu-iinn liii 



Further, if lt-> n -\> n -< nt i! limi cocHirifiit r* 7 * in tin urn /,'.' shall d- 



2 " 



I In- valm- of /'., mrnsiirvil from its mean value p =? . 



-i 

 The correlation surfan- of/// and ft, is a somewhat r,m|.li<-ai,. -I >n- namely 



. (xxxiv). 



II \vc integrate out for y x we obtain, with proper interchange of letters, the same 



V 



curve of distribution for RZ' as we obtain in equation (xxiii) for // = .R, - p ~ ' 

 For JRg' constant, we see that the distribution of y x ' is a normal curve with 



Mean y x ' = R t '(^-m 1 ) ........................ (xxxv), 



or there is linear regression. But the standard deviation of y x f for constant RJ is 

 given by _ 



s,vrv/ /y. \i 



B' ff iix' = - /= -- f ! + T2 - I ............... (XXXVl). 



vn I ? n - ^ / 



V 2^ (1 r>J 



In other words the distribution of R^'^yx' ^ s heteroscedastic, and the scedastic curve 

 is part of a hyperbola. 



The distribution of y x ' depends upon the integration with regard to RJ of 

 equation (xxxiv), and this has not yet been achieved in any simple form*. But we 

 can obtain the constants of the frequency curve for y x . 



The curve for y x ' is symmetrical, or the 



Mean y x =0 .............................. (xxxvii). 



2 /*-i\) 



................................. (xxxix), 



or all odd moments vanish. 



2/ 2 





, 2 . - , 2 . fx-m 1 \ z 



where or is written tor 1 h I ^ 1 . 



n \ 2.1 I 



* It depends on expressing in some simple form or series the integral 



y -i/ 



where R% = ~ \/l - p 3 tan demies 0, and a = >/n y x '/^-2 "^1 - P* an( i ^ = v** ( x ~ 

 B. II. 



