128 MAJOR P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



we obtain 



and since XT XT XT,. t / UXT 



N f .,- = N,,_,_e +lil '-<, N;;i, +I = (-)'N,, 



and 



/N m WN ro y'_. 

 we obtain ^, 



_ /N n _,, t+1 WN n _ M+l y" 

 ' ( P> ) ( P, ) 



Art. 110. We have two alternative expressions for 



in terms of numbers 



I verify them in the case 



J 



+ 3(N m -N m _ 1 ) 15 



agreeing with, for m = 3, 



8.48 = 302 + 3(66-26) + 3(ll-2.11 + l) + (l-3. 4 + 3.1). 



N Mi! ,= 



and, for m = 3, 



= N 4>1 . . 



-3(N 4 -N 3 ) 1S 

 + 3(N 4 -2N,+N,) 1 . 



agreeing with 



8.48 = 302-3(26-66) + 3(l-2.11 + ll)-(-3.1 + 3.4-l). 



