MAJOR P. A. MAC-MAHON ON THE COMPOSITIONS OF NUMBERS. 109 



Art. 67. Again, operating with 





m 



we obtain a result -of the form 



A (/<,-o,)+ B (Mi -o/i) 



(A + B){(3) + (21)}; 



hence ^.-^ (/,_) = D.-D. (* 



equivalent to 



and, particularly, from the tahle 



N (321) (W -N (2*1 V) = N (321)<,-N (2'!%,,, 

 8 3 = 20 15. 



No new result is obtained by taking 



as the operand. 



Art. 68. Further, Di"*(V. - 



has the form 

 reducing to 



equivalent to the new result 



and, particularly, from the table 



18 13 11 - 6. 



Art. 69. If we take here the operand to be 



a new result is obtained, viz., 



The above is sufficient to indicate the nature of the results which present them- 

 selves ; I have not attempted to generalise them. The question appears to be a 

 difficult one. 



