692 



SCIENCE, 



[N. S. Vol. XXII. No. 570. 



resolves itself into this one : What is— what 

 must be — the ratio of immigrants to natives 

 within any area ? What factors determine 

 that ratio? 



Ceteris paribus, one factor must be the 

 size of the area. If the area be a small one, 

 say of ten yards radius, suppression of 

 propagation will do little good, because the 

 proportion of mosquitoes bred there will be 

 very small (under our assumed conditions) 

 compared with those which are bred in the 

 large surrounding tracts of country, and 

 which will have no difficulty in traversing 

 so small a distance as ten yards. But if 

 we completely suppress propagation over 

 an area of ten miles radius, the case must 

 be very different— every gnat reaching the 

 center must now traverse ten miles to do so. 

 And if we increase the radius of the no- 

 propagation area still further, we must 

 finally arrive at a state of affairs when no 

 mosquitoes at all can reach the center, and 

 when, therefore, that center must be abso- 

 lutely free from them. In other words, we 

 can reduce the mosquito density at any 

 point by arresting propagation over a suf- 

 ficient radius around that point. 



But we now enter upon more difficult 

 ground. How large must that radius be 

 in order to render the center entirely mos- 

 quito-free ? Still further, what will be the 

 proportion of mosquito reduction depend- 

 ing upon a given radius of anti-propagation 

 operations ? What will be that proportion, 

 either at the center of operations, or at any 

 point within or without the circumference 

 of operations? The answer depends upon 

 the distance which a mosquito can traverse, 

 not during a single flight, but during its 

 whole life; and also upon certain laws of 

 probability, which must govern its wander- 

 ings to and fro upon the face of the earth. 

 Let me endeavor to indicate how this prob- 

 lem, which is essentially a mathematical 

 one of considerable interest, can be solved. 



Suppose that a mosquito is born at a 

 given point, and that during its life it wan- 

 ders about, to and fro, to left or to right, 

 where it wills, in search of food or of 

 mating, over a country which is uniformly 

 attractive and favorable to it. After a 

 time it will die. What are the probabili- 

 ties that its dead body will be found at a 

 given distance from its birthplace? That 

 is really the problem which governs the 

 whole of this great subject of the prophy- 

 laxis of malaria. It is a problem which 

 applies to any living unit. We may word 

 it otherwise, thus— suppose a box contain- 

 ing a million gnats were to be opened in 

 the center of a large plain, and that the 

 insects were allowed to wander freely in all 

 directions — how many of them would be 

 found after death at a given distance from 

 the place where the box was opened? . Or 

 we may suppose without modifying the 

 nature of the problem that the insects 

 emanate, not from a box, but from a single 

 breeding pool. 



Now what would happen is as follows: 

 We may divide the career of each insect 

 into an arbitrary number of successive 

 periods or stages, say of one minute's dura- 

 tion each. During the first minute most of 

 the insects would fly towards every point 

 of the compass. At the end of the minute 

 a few might fly straight on and a few 

 straight back, while the rest would travel 

 at various angles to the right or left. At 

 the end of the second minute the same 

 thing would occur- most would change 

 their course and a very few might wander 

 straight on (provided that no special at- 

 traction exists for them). So also at the 

 end of each stage — the same laws of chance 

 would govern their movements. At last, 

 after their death, it would be found that 

 an extremely small proportion of the in- 

 sects have moved continuously in one direc- 

 tion, and that the vast majority of them 



