LOGICAL BASIS OF MOSQUITO-REDUCTION 93 



depending upon a given radius of anti-propagation operations? 

 What will be that proportion, either at the centre of operations, 

 or at any point within or without the circumference of operations? 

 The answer depends upon the distance which a mosquito can tra- 

 verse, 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 problem, which is essentially a mathematical 

 one of considerable interest, can be solved. 



Suppose that a mosquito is born at a given point, and that dur- 

 ing its life it wanders 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 probabilities 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 prophylaxis 

 of malaria. It is a problem which applies to any living unit. We 

 may word it otherwise, thus suppose a box containing a million 

 gnats were to be opened in the centre 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 ca- 

 reer of each insect into an arbitrary number of successive periods 

 or stages, say of one minute's duration each. During the first min- 

 ute most of the insects would fly towards every point of the com- 

 pass. 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 attraction ex- 

 ists 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 direction, and that the vast 

 majority of them have wandered more or less backward and for- 

 ward and have died in the vicinity of the box or pool from which 

 they originally came. 



The full mathematical analysis determining the question is of 

 some complexity; and I cannot here deal with it in its entirety. 

 But if we consider the lateral movements as tending to neutralize 

 themselves, the problem becomes a simple one, well known in the 

 calculus of probabilities and affording a rough approximation to 



