652 
MAJOR P. A. MACMAHON ON THE 
This representation is only possible when the subjacent succession of lines is insisted 
upon. 
Art. 44. Every Sylvester-graph in two dimensions is representative of two 
unipartite partitions ; it may, in fact, be lead by lines or by columns, and when the 
two readings are identical the graph is said to be self conjugate. 
In this enlarged theory every graph denotes 3 ! graphically-regularised multi¬ 
partite partitions; of the same total content, but not, as a rule, appertaining to the 
same multipartite number. 
Take coordinate axes as shown, the axis of z being perpendicular to the plane of 
the paper. We read as follows :— 
Planes parallel to the plane xy and in direction Ox 
(643 632 4U). 
Planes parallel to plane xy and in direction O y 
(333211 332111 311100). 
Planes parallel to plane yz and in direction O y 
(333 331 321 211 110 1L0). 
Planes parallel to plane yz and in direction Oz 
(333 '322 32l 3Tb 200 200). 
Planes parallel to plane zx and in direction Oz 
(333322 322lb0 321000). 
Planes parallel to plane zx and in direction Ox 
(664 43l 32l), 
the multipartite numbers, of which these are partitions, being 
(16, 8, 6), 
(976422), content 
(T37TT76), 30 
(T6T876), 
(976422), 
(13, 11, 6). 
