74 MAJOR P. A. MxcMAHON ON THE COMPOSITIONS OF NUMBERS. 



Now, combining the two sets of numbers, we find that either there is or there is not 

 a break in the descending order between 



a, and ,+, ; 

 hence the number of arrangements is also 



.rt s+( ) + N( 1 a 2 ...a.-i, a,+a s+l , a I+2 ..., +( ). Q.E.D. 



Art. 13. Regarded as a numerical theorem, the multiplication is commutative, but 

 in regard to form it is not commutative ; thus, by considering the multiplication 



(a, +1 a, + ,...e^+) N (!.,...,), 

 we obtain the linear relation 



Observe also that the order of the numbers in brackets in any number N(...) can 

 be reversed at pleasure and thus new forms of results obtained. 

 As a verification : from the tables 



N (12) N (11) = N (121 2 ) H- N (131) = N (1 3 2) + N (12 2 ) ; 

 10 . 2 . 1 9+11 4+16 



N (123) + N (15) = N (312) + N (42). 

 35 + 5 = 26+14 



The fact that the multiplication is not commutative formally is of great importance 

 in the theory of these numbers. 



Art. 14. Extending the theorem to the product of three numbers 



N (!. . .a.), N (&A. ..b t ), N (cjC.,. . .c u ), 

 we find 



' 



+ N(aj. ...&! &<-!, ^t + ^l, C 2 ...C U ) + N(i-. .,-!, .+ />!, /> 2 --.^-l, ^ + Ci, C 2 ...C U ). 



We may, in general, give the right-hand side 3 ! different forms corresponding to 

 the 31 permutations of the numbers N(...) on the sinister. 



If we take the product of m numbers N (...), to form the dexter, we combine the last 

 integer of a number N (...) with the first integer of the next following number N (...), 



