42 KARL PEAESON 



If we make b infinite, then P,(e 1 ) and P s (e 2 ) = for s>0, and A( c i) an d 

 A(e 2 ) = 0, P (e 1 ) = Po(e 2 )=i, and__ 



f n (h, k) = N{l-P ( Vl )-Pt{ v ,)-L l (r ll )-L,( Vl )} (lxxiii). 



This could be deduced directly from (xlix) and it represents the first migration 

 distribution into an indefinitely long cleared strip or belt. This is a result of 

 some interest as it might approximately apply to the migration into a zone cleared 

 by a flood or a fire of certain types of animal or vegetable life. 



(13) Problem VI. To determine the distribution after m migrations into a 

 cleared but not sterile rectangular area. 



By the general proposition on p. 39 we have only to write "% = m<j for cr, 

 and the iV's for the v's in the L's. Let us put 



r) 1 ' = (a-h)/'Z = r) 1 /Jm, V = W«M e / = e i/7 TO > ej = ejjin. 

 Let 



A' M = 1 4= v^~ W \N^ ( Vl ') + N& ( % ') + . . . + N^-v (V) + ».}. 



2 V27T 



and so forth, then we have for the full solution : 

 m F n (h, k) = (/tA)"" W[{P (V) + Po (V)} {Po (O + Po (*')} 



+ {A (>?/) + A' ( V)} {P (€,') + P (e/)} + {P ( V) + P (V)} {A' (O + A' (*')} 



+ {P, (e/) + P, («/)} {A M + A' (V)} 



+ . . . + {P. (e/) + P s (e/)} {Z7. +1 (V) + L' s+1 (r,;)} + . . .] (lxxiv). 



The terms here will very rapidly converge for any fairly large value of m, 

 so that for many purposes we may write the solution : 



m F n (h, k) = (/xA)- W {P (V) + P„ (V)} { A> (O + Po («.')} ( lxxv )> 



which can be found at once from the usual tables of the probability integral. 



Illustration I. A rectangular patch 2 miles long and 1 mile broad is cleared of 

 mosquitoes, but not retained sterile. What would be the central density at the 

 end of the year? Suppose 10 breeding cycles with their scatter migrations, , each 

 of 6 flights, to take place in the year. Then if we take 200 yards as a possible 

 round value for the flight we have : 



m = 10, n = 6, Z = 200yds., cr = ^Z 2 = 120,000 or cr = 346-41 yds. 

 a = 880 yds., 6= 1760 yds., rj, = i) t = 2-540, e x =e 2 =5-081, 



2 = Vf0o-= 1095-44 yds., V = V= ' 803 > e/ = e/= 1-607. 



Hence 10 P 6 (0, 0) = GaA) 9 4P (-803) P (1-607) N, 



or, using Sheppard's Tables : 



10 P 6 (0, 0) = (/aA) 9 4 x '2890 x "4460iV, 

 = (jaA) 9 x-5156iV. 



