MAJOR P. A. MACMAHON ON THE CO^OSITIONS OF NUMBERS. 93 



formed according to the same law, but the successive blocks of terms having 

 alternately positive and negative signs. 



Art. 41. I proceed to generalise the two results 



By definition 



6 V {(,... a._,a.) (6,6,... 6,)} 



j.-.&^ + N (a,.. 

 and since 



this 



Now the sum of these two terms is precisely 



because the terms involving , 



are the same, with opposite sign, as those involved in 



and therefore cancel them. 

 Hence the result 



Art. 42. Again 



t _ 1 , a. 



by successive use of the formula Art. 4 1 above. 



