622 
MAJOR P. A. MACMAHON ON THE 
Art. 5. The separations of a given partition may be grouped in a manner which is 
independent of their specifications. 
Consider the separable partition 
(PiPz)> 
which is itself to be regarded as one amongst its own separations. 
Viewed thus, it has qud partition a multiplicity (32). 
Write down any one of its separations, say 
(Pi) (PiPz) (Pi)- 
This separation may be regarded as being compounded of the two separations 
(. Pi) ( Pi ) of M 
and 
( P-2 ) 2 Of ( Pi)' 
Three other separations enjoy the same property, viz., 
{Pi) (lb) (PaY> 
( P\P 2 ) (Pi) (Pi)> 
(PiPz) (PiPi)l 
for, on suppressing p 2 in each, we are left with 
and on suppressing p l there remains 
These four separations 
(pi 2 )(pi); 
(p 2 ) 3 . 
( Pi) ( P1P2) (Pi)~) 
( Pi 2 ) (Pi) {pJ 2 * 
{PiPz) (Pi ) (Pa) ( 
(Pi 2 P 2 ) (Pi Pa) J 
1-Set {(21), (l 2 )}, 
form a set which is defined by two partitions, one appertaining to each of the two 
numbers which define the multiplicity of the separable partition. 
Thus the first number 3 of the multiplicity occurs in each separation of the group 
in the partition (21), and the second number 2 in the partition (l 2 ). 
There are as many sets of separations as there are combinations of a partition of 3 
with a partition of 2. 
