394 



Prof. Kar] Pearson. 



-^i + 7/3 + 7/3Vi + 7/^+ .... +lP n Pn-i = 

 -/> 2 + 7fe + 7/3 3 + 7)8Vi+ +lft n Pn-2 = 



0. 

 0. 



~Pq + ifiPq-l + 7/ 3 V?-2 + lft 3 Pq-3 + • • • • + 70V»-J = °- 

 — / J 2+l + 7/ 5 / > 'Z + 7^V?-l + 7/ 3 V?-2+ .«•• + VfiPpn-q-l = 0. 



— />* 4- 7^»-i + 7/ 3 V«-2 + 7/ 3 V«-3 + . . . . + 7/3" = °- 



Multiply the ^-flth equation by 1//3, and subtract from the gthj 

 we have 



|^+i-p 2 (i4-7) + 7^>-g = ( xi )> 



Assume ^ = ca?, hence : 



^-(l + 7) + 7)3V 2 2 = 0. 



But since a and /3 are both less than unity, the last term will be 

 vanishingly small when n is indefinitely large, thus : 



* = /3(l + 7 ) (xii). 



Substituting p q = c«0 in the first of the equations for the jo's above,, 

 we have : 



Or, taking as before (ccfi) 11 = for n very large : 



1 — /3a, 



1 fl (l + 2 7 ) 



y0 P 7 



(xiii).. 



(xii) and (xiii) contain a complete solution of the fundamental 

 equations for the />'s given above, so long as we go only to a finite 

 number of mid-parents, i.e., q may be very large, but not comparable 

 with n == oc . 



(6) Special Cases. 



(a) Put 7 — . 1, ft = J. It follows that a = 1, and c = 1. Hence if 



_ l 



