30 KARL PEARSON 



(9) Problem II. To investigate the distribution after m migrations from uni- 

 formly densely occupied space across a straight boundary into unoccupied space. 



Let the axis of x be taken perpendicular to the boundary and the axis of 

 y be the boundary. Let us consider the density at x = c, on the originally 

 unoccupied side of the boundary. Then the density at a distance x from the 

 boundary is given by (xlvii), or if we write the operator as Q t , we have 



F n {x) = NQ t ^ r \ e ^dx = Ul , say, (li). 



V ZlT J x\a 



Here Q t involves only n and cr and not x. 



Now the distance r from the point x, y to the point c, at which we 

 want the density after the next migration is given by : 



r* = y 2 + (x — cf, 

 and jliA being the fertility-survival factor, we have for the density at c, 



u 2 — I (jbAu^n (r 2 ) dxdy. 



J — 00 J —oo 



Now <t> n (r°) = Q t <o = Q t ^ e-^°\ 



To mark that this Q t operates only on this part of the expression, write it 

 Qt and suppose it to operate on cr' written for cr. After the operations are com- 

 plete we can put cr' again =cr. Let 



-^dx. 



J2tt J xh 



Then if juA be constant (see p. 29) : 



u > ^ 



Completing the integration with regard to y we have : 

 Differentiate with regard to c : 



Integrate by parts, and notice that the part between limits vanishes at both 

 of them and we have : 



P = i^£ Q tQ > r p i, e -t<-->V dx . 



dc J2tt J -oo dx cr' 



But ^ = 1 Ie-*CA*; 



dx J2ir o" 



hence: ~3 1 = -^^QtQt —,\ e ^ " 2 ' dx. 



ac Air crcr J _<„ 



