376 MR. S. M. JACOB ON 



r = 



VV+W+V) vV+Wy'+V) 



since the original correlation was perfect. 



This is sufficient to indicate the method, and we will now turn to the general 

 case of a correlation less than unity. 



The original correlation being assumed to be normal the standard deviation of 



each y array is <yv/i -f' 2 - Then the frequency of a character of deviation y s from the 

 mean of the array corresponding to x m is given by 



N m ~ a- „% 



ys" 



W2*(i-r*) 



^(i-*)aj,,-£8y -f 



say where &y s is a small interval enclosing y s . 



Now owing to inaccurate measurement of the assumed type each character in the 

 interval %y s gets a different value, so that the whole group is distributed about the 

 mean character y s with a standard deviation k (ym-\-y s ). 



Thus the standard deviation of the whole array corresponding to x m is altered 

 to a value <**„' where 



N, 



•+oo 



°" y - /t "/-« ps w+Wy^+y*) 1 *y* 



=iV w «r ym a +\ 7=* ym {ym+y,}Vy 





J-on'W 



00 • " " - 2B 





=N m (<r ym l +kJ m *)-\-- — 7=1 « ffy " (2ymys+yV) <*y,. 

 Now the first part of the integral must clearly vanish, and we require to find only 



y- i 



S+oo -\ — ; i f + co (l 



e ^ysUysA -°yjys- 

 — 00 J —00 





s 

 s 



dy s 



i y. 4 , i y. % 



t— I a-, +00 f+00 -2 a 



s 



oo J -oo 



= v ym \ Nm^tt a ym 



= V^a"- N m °yj 



