MAJOR P. A. MAcMAHOX ON THE COMPOSITIONS OF NUMBERS. 75 



times iii 1 way , 



/T-l\ 



f l j ways, 



hence 2*" 1 numbers N (...) present themselves on the dexter. 



Not counting reversals of order, the dexter can, in general, be given as many 

 different forms as there are permutations of the numbers N(...) on the sinister. 

 Counting reversals, the number of different forms is further multiplied by 2", subject 

 to a diminution when one or more of the numbers N (...) is self-inverse. 



Applications of the Theorem. 



Art. 15. The theorems, already arrived at above, are particular cases of multipli- 

 cation. Thus the formuhe, of which 



N (abc) + N (a + 6, <) + N (a, b + c) + N (a + 6 + r) = -^Lj 

 is a type, are equivalent to results, of which 



j^Tj N () N (6) N (c) = N (abc) + N (0+6, r) + N (a, b + c) + N ( + b+c) 

 is representative, since j^ / fl \ = j^ m _ N (c) = 1. 



That the sum of all numbers N (...), of given weight n, is >* ! is shown by the formula 



since on the dexter occurs an N(...) corresponding to every composition of n. 



Art. 16. Suppose that it is required to find the sum of all numbers N (...), of given 

 weight, which are such that each associated composition commences with a given 



series of numbers 



a l a t ...a m , 



or, in other words, suppose we wish to make the summation indicated by 



the solution is given at once by 



for, by the multiplication process, the unit which terminates N (n l a,...a m \), combined 

 with "-* 1 - 1 



