Bramble-bees and Others 



positions of this first Osmla can be combined 

 each of the two positions of the second, giving 

 us, in all, 2X2 = 2- arrangements. Each 

 of these 2- arrangements can be combined, in 

 its turn, with each of the two positions of the 

 third Osmia. We thus obtain 2X2X2 = 2^ 

 arrangements with three Osmias; and so on, 

 each additional insect multiplying the previous 

 result by the factor 2. With ;; Osmise, there- 

 fore, the total number of arrangements is 2". 



But note that these arrangements are sym- 

 metrical, two by two: a given arrangement to- 

 wards the right corresponds with a similar ar- 

 rangement towards the left; and this sym- 

 metry implies equality, for, in the problem in 

 hand, it is a matter of indifference whether a 

 fixed arrangement corresponds with the 

 right or left of the tube. The previous num- 

 ber, therefore, must be divided by 2. Thus, 

 n Osmia?, according as each of them turns her 

 head to the right or left in my horizontal tube, 

 are able to adopt 2"'^ arrangements. If n = 

 10, as in my first experiment, the number of 

 arrangements becomes 2° = 512. 



Consequently, out of 512 ways which my 

 ten insects can adopt for their outgoing posi- 

 tion, there resulted one of those in which the 



43 



