124 MAJOE P. A. MAcMAHON ON THE COMPOSITIONS OF NUMBERS. 



In every case, if <r be the highest figure in the perfect partition, D,, is a factor of 

 the expression for D,, e.g., taking the perfect partition 



3*l 2 , 

 40D 6 = 2D 3 2 +3&D S D 1 2 +15& 8 D 3 D 1 + 20?/D S . 



I do not interrupt the investigation by stopping to prove the theory of expressi- 

 bility depending upon perfect partitions ; its truth is intuitive. 



Art. 97. It is necessary to labour the subject of the operator relations, qud the 

 special operands, because the whole theory of the numbers N ra is involved. 



Art. 9. Perhaps the most interesting of the operator relations are those which do 

 not involve b (or X). 



Recalling the relation of Art. 92, viz., 



where _, _, 



ztrp = n. ZTT = t, 



we may also write 



J n 



where 



and, if 



n-i = v-j, 



we may eliminate b, obtaining 



Art. 99. The simplest formula thence obtained is found by putting 



and this leads at once to , , 7 , 



rijrfs-rt/ = 0, 



D 2 D 1 2 -4D/+3D 3 D 1 = 0, 

 which also results by elimination of b from 



Art. 100. To obtain spme more relations in a simple manner, I write 



