88 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
301, 130, 103, 031, 013; 220, 202, 022; 211, 121, 112, the law of operation is as 
follows :—- 
DAW = (DAP.Q.)(W+( WA)(DA)+(D,Q.)(DA)(DA) 
+ (DA)(D,Q 1 )(D 0 Q c ) + (D 3 <U(DA)(D 1 Cy + (D,Q a )(D,Q,)(D„Q t ) 
+ (D 1 Q,)(D, 1 Q S )(DA) + (D 11 Q„)(D 3 a i )(D 1 Q e ) + (D 0 <y(D,Q s )(DA) 
+ (H2Q,;)(IlaQ, ft )(D l ,Q e ) + (D 2 Q a )(D 0 Q f/ )(D 2 Q ( .) + (D 0 Q a )(D 2 Q, f) )(D2Q, ( .) 
+ (D 2 Q a )(D 1 Q i )(D A) + (D 1 Q a )(D 2 Q 6 )(D 1 Q c ) + (D ] Q a )(D 1 Q t )(D 2 Q e ) 
D 0 , being unity, may be omitted but has been retained above to make the 
connexion with the compositions quite clear. This method of performing D m upon 
a product I have explained and used in previous papers during the last five and 
twenty years. _ Upon this example some observations can be made. In the first place 
the operation breaks up into 15 portions because the number 4 has 15 three-part 
compositions. The result of each portion must be moreover either Q a Q 6 Q c or zero, 
because D ; Q S is either Q s or zero. Hence the result of the whole operation must be 
merely to multiply by some integer ^15. In general the result of the 
operation 
DJW...Q* 
must be merely to multiply the product 
by some integer equal to, or less, than the number of compositions of m into 
k l + k 2 +... +Jc { parts, zero always counting as a part. 
Hence also the result of the operation 
D mi D m2 ... D^Q^'Q,*’ 
must be merely to multiply the product 
by an integer. 
This valuable result shows that the Hammond operators may be performed with 
facility upon the function 
F *w 
which is before us. 
Art 7. The determination of the result of the operation 
is now entered upon. We have to find the value of the multiplying integer 
