398 Prof. Karl Pearson. 



Now make use of the general equations for the p's given just below 

 equation (x), substituting for the R's in terms of the e's, and remem- 

 bering that we are to stop at n = q, that p p = cocP, and that 1/^/2 = a. 

 IV e have after some reductions the system : 



1 == ^e l!Z + ^ 2 6 22 + 5; 4 e3 !? + a G e4 ? + + ^ q ~ 2e qq, 



1 = e iq + ~ c%q + a?c H + cc i e iq + .... +a 2a_4 6 ?2 , 

 G 



1 — eiq + e 2 q + - e 3q + X?e iq + + cJ 9 ~ 6 e qq , 



1 = e 12 + e 22 + e 32 + .... -f-6 ? _ lf + a 2 e ?? , 



1 = e iq + 6 22 "I" e 3q + • • • • + <^ _ x 2 + ~ 



Subtracting the ((7 — l)th of these equations from the qt\ the 

 (9 — 2)th from the (q — l)th, &c, and introducing the values of 

 a 2 = ^ and of c = 0*6, we find 



0-4 c 2 _! ff — 0*7 e 22 = 0. 



0'4 e ? _ a 2 — 0*7 6 ? _i 2 — 0*15 e qq = (xv). 



0'4 6 ? _ 3 2 — 0*7 6 2 _ 3 q — 0'lS 6 2 _x ? — 0"075 = 0. 



0-4 6 2 _ 4 ? — 0*7 6 2 _ 3 2 — 0"15 e 2 _ a q — 0-075 2 — 0-375 e 2? = 0. 



and so on, each new coefficient being now half the last. These equa- 

 tions give successively the ratios of e q _ l ? , e q _ 2 q ,e q _ 3 q , &c, to e qq . 

 Hence the last of the previous set of equations will then give e qq . 

 Thus the partial regression coefficients for any limited number of 

 mid-parents can be found. This last equation also gives us 



o/ \ 1 — c 2 

 fe( 6 ) = 1 — e qq = 1 — - e qq , 



a convenient formula for measuring how nearly the mean offspring 

 of q mid-parents, all selected with a peculiar character, h L = k 2 = h z 

 = . . . . = k q = K has attained that character. For in this case 



lc =S(ek) = KxS(e), 

 2 



•and lc /K — 1 — - c qq (xvi). 



