Mathematical Contributions to the Theory of Evolution. 145 



parent, there will be p^ya^^ like the grandparent, and so on. But 

 beyond these contributions, certain of the who follow the parent 

 will be like the grandparent, for the parent is like the grandparent 



in pi fraction of cases. 

 Hence we have finally : 



p2^ = Pi^N + yaN + 2/o^7a'N + 4/)27a3N + 



or p2 = piP + ya + 2ycc\pi + 2a/)2 + ia?pz + ) (iii) 



Proceeding in the same manner we find 



pz = P2P + yoipi + yoL^ + 2yoi\pi + 2a/02 + ia?pz + ) (iv) 



Pi = psP + yap2 + yoL^pi + ycc^ + ^yo^\p\ + 2a/02 + ia^pz + . . .) . . . (v) 



and so on. 



Hence we deduce from (iii) and (ii) 



P2 = PiP + yoc + a{pi- 



or ' p2 = + /5)/oi + a(7 - /3) (vi) 



Similarly from (iv) and (iii) 



Pz = P2^ + ym + ^(P2-piP) 



or P3 = {cc + /i)p2 + cc(y-fi)pi (vii) 



Again from (v) and (iv) 



P4 = ps/^ + 7a/)2 + a(p3 - P2/5) 



or P4 = (a + iS)p3 + a(7 - /5)/)2 (viii) 



Generally pn = + P)pn-i + oc(y - P)pn-^ (ix) 



with Pq = 1, by (vi). 



To solve equation (ix) assume as usual pn = Am'\ and we find 

 m2 - (a + /3)m + a{y - fS) = 0. 



Thus m = ^ + ^± Ji^ + Py + Hy-f^) 



2 



But by (i) 1 - 2(a + ^) = 4a(7 - 



Hence = 1 or m = a + ^- J (x) 



We have then 



where Ai and A2 are constants. 



