2IO APPE;NDIX II — INTENSIVE SBGREGATION. 



digesting the same, so that, in each district alike, one in a miUion could 

 survive in this way though the crop of leaves should fail. 



(4) Suppose that there are, through diversity of adaptations of this 

 kind to products of the environment, ten different kinds of accessible 

 forms of food, on each kind of which one in a million of the individuals 

 of each district might feed if driven by necessity. 



(5) Now suppose the same necessity should occur in each district 

 through the destruction of the leaves on which they habitually feed, 

 and that there are accordingly in each district a hundred survivors 

 able to maintain themselves on other kinds of food. 



Under such circumstances (the correspondences of which we have in 

 our supposition made much more exact than the actual deviations from 

 a mean ever present), even under such circumstances of completely 

 parallel variation, what is the probability that in each of the separate 

 districts the few that would meet with other individuals and have an 

 opportunity to propagate the species would be similarly endowed and 

 similarly related to the environment? 



In order to still further simplify the problem, let us assume that in 

 the case of each kind in each district the probability that it will suc- 

 ceed in propagating is exactly balanced by the probability that it will 

 fail. The probability, then, that any given number, o, of the ten kinds 

 in a given district will succeed is found by estimating the number of 

 ways in which a things can be taken out of 10 things, and dividing 

 this number by the tenth power of 2, that is, by 1024. This is com- 

 pletely parallel to the number of ways in which ten pennies can be 

 arranged as to head and tail, each penny representing one form of varia- 

 tion, and its lying head-up indicating success in propagating. In 1024 

 experiments the probability is — 



That 6 will succeed 210 times 



That 7 will succeed 1 20 times 



That 8 will succeed 45 times 



That 9 will succeed 10 times 



That 10 will succeed i time 



That o will succeed i time 



That I will succeed 10 times 



That 2 will succeed 45 times 



That 3 will succeed 120 times 



That 4 will succeed 210 times 



That 5 will succeed 252 times 



These figures are found in the eleventh line of what is known as the 

 Table of the Binomial Coefficients, " or the " Arithmetical Triangle. ' ' * 

 And so in the case of any number of objects, the number of combina- 

 tions that may be made with n objects is found in the w + ith line of 

 the arithmetical triangle classified according as there are o, i, 2, 3, or 

 more objects in each combination. The whole number of combina- 

 tions may also be found by calculating the wth power of 2. 



* See "Principles of Science," by W S. Jevons. 



