MAJOR P. A. MA.MAHOX ON THE COMPOSITIONS OF NUMBERS. 115 



* 



and, repeating the summation 6 times, 



so that, when 6 = n, 



Again, taking differences instead of summing, we get the series 



XT \r c /n+2\ r /n+2\ r 



JN m -JN M _, = ^~( ! )<-s,-i+( 2 JL_t 



N-2N. 



and in general 



These restilts are all given by the two formulae 



which become the same when p = 0. 



Curious Expression for 

 Art. 75. I shall now prove that 



N _p m-2/7t\ m-3/u\ 



N - p --z 



where 



P.- ^- 



o 



denotes the expansion of 



Q 2 



