98 PREVENTIVE MEDICINE 



be exactly lost by the undrained country. This fact can be seen to 

 be obviously true if we imagine an immense mosquito-bar put up 

 along the line of demarkation so as to check all migration across it, 

 when, of course, the mosquito-density would remain as at first on 

 the left, and would become absolute zero on the right: then on re- 

 moving the mosquito-bar an overflow would commence from left 

 to right, which would increase the density on the right by exactly 

 as much as it would reduce the density on the left. 



The dotted line on the diagram indicates the effect on the mos- 

 quito-density which must be produced by the drainage. If L is the 

 possible limit of migration of mosquitoes (it may be one mile or a 

 hundred, for all we know), the effect of the drainage will first begin 

 to be felt at that distance to the left of the boundary-line. From this 

 point the density will begin to fall gradually until the boundary is 

 reached, when it must be exactly one half the original density. This 

 follows because of the equivalence of the emigration and immigration 

 on the two sides. Next, as we proceed from the boundary into the 

 drained country, the density continues to fall, until at a distance 

 L on the right of the line, it becomes zero, the country now becom- 

 ing entirely free of mosquitoes because they can no longer penetrate 

 so far from the undrained country. 



In the diagram the line giving the mosquito-density falls very 

 slowly at first, and then, near the boundary, very rapidly, subse- 

 quently sinking slowly to zero. The mathematical analysis on which 

 this curve is based is too complex to be given here ; but it is not diffi- 

 cult to see that the centripetal law of random migration must deter- 

 mine some such curvature. The mosquitoes which are bred in the 

 pools lying along the boundary-line must remain for the most part 

 in its proximity, only a few finding their way further into the drained 

 country, and only a very few reaching, or nearly reaching, the limit 

 of migration. Though an infinitesimal proportion of them may wan- 

 der as far as ten, twenty, or more miles into the drained country (and 

 we do not know exactly how far they may not occasionally wander) 

 the vast bulk of the immigrants must remain comparatively close 

 to the boundary. And as, for the reason just given, the mosquito- 

 density on the boundary itself must always be only one half the 

 original density, it follows that it must become very rapidly still 

 less, the further we proceed into the drained country. In fact, the 

 analysis shows that the total number of emigrants must be insig- 

 nificant when compared with the number of insects which remain 

 behind that is, when they are not drawn particularly in one direc- 

 tion. We are, therefore, justified in concluding that, as a general 

 rule, the number of immigrants into any area of operations must, for 

 practical purposes, be very small or inappreciable a short distance 

 within the boundary-line. The following diagram probably repre- 



