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APPENDIX II — INTENSIVE SEGREGATION. 



A method of constructing the perinutiiional triangle from the arithmet- 

 ical triangle. — Pascal's arithmetical triangle, which is the same as the 

 table of binomial coefficients, is a series of figures, each line of which 

 may be formed by adding the previous line to itself, as shown in the 

 table below. Now, if we compare this arithmetical triangle with my 

 permutational triangle we find that the first and third diagonal lines 

 in each table are composed of the same numbers arranged in the same 

 way. The fourth diagonal line of the permutational triangle can be 

 obtained by multiplying each number of the arithmetical triangle by 2 • 



1 



Multipliers 



In short, by using the numbers here indicated as multiphers, each 

 line of the arithmetical triangle may be transformed into the corre- 

 sponding line of the permutational triangle. It may further be noted 

 that these numbers by which we multiply are the occurrents standing 

 in the first column of the permutational triangle; and these are found 

 to be the same as the sub-factorials described by Whitworth in 

 "Choice and Chance," Chapter IV. 



