276 Prof. K. Pearson. On a Criterion which may [Mar. 4, 



/*22 = (^22 - p) % + (m 2 - 111) (1 - f 22 ) 



= (ru ^ - a; + (mi - m) (1 - r i2 ). 



Hence, generally for the variability of the brethren of a given brother 

 of deviation x, we have 



9 Vi - 1 ,o V 2 - 1 . o v 2 

 + j"ir ^77 -V, + 22 ^77 77 + ^ .77 77 



2(vi + v 2 -l) 2(v 1 + v 2 -l) 2(vi + v 2 -l) 



+ ^21 2 Q/ ^ Tx ( Xix )« 



2 (vi + v 2 - 1) 



Clearly this gives a hyperbola for ~2 X in terms of x, with its real 

 axis perpendicular to x. 

 Similarly we have : 



p« 2 = jL v \ ( ; i ~ 1) n hi^+(^ 



(vi + v 2 ) ("1 + v 2 - 1) 



+ 



~~W T\ '! r 22°"2 + («?2 - mf\ 



(v 1 + v 2 )(vi + v 2 - 1) 



+ ; 77 {? 120-10-2 + (mi - m) (m 2 - m)} (xx), 



(vi + v 2 ) (vi + v 2 - 1.) 



and further 



0-2 = — ^ W» + (mi - mf] + {<r 2 2 + (m 2 - m) 2 } , 



V X + V 2 Vi + V 2 



, v Y m\ + v 2 m 2 



and m = — — 1 — . 



mi + m 2 



In any actual case we may reasonably suppose our pairs of brothers 

 to be a random sample from families with indefinitely great numbers,* 

 or put vi and v- 2 infinite in the ratio of % to n 2 . Hence : 



m = (mini + m 2 n 2 )/n , a- 2 = (nicri 2 + n 2 ar 2 2 )/n + 7 ^^(mi - m 2 ) 2 , 



n 2 



o ni 2 o , no 2 9 , 2n x n 2 /™\ 

 po- 2 = -i- riia-i 2 + r 22 o- 2 2 + —\— ri 2 (ria- 2 (xxi), 



* < Phil. Trans.,' A, vol. 203, p. 77. 



