650 
MAJOR P. A. MACMAHON ON THE 
or lie on the side of it towards J. Such lines of route may be termed inferior 
or subjacent to the given line of route. Similarly those lines of route which, every¬ 
where, are either coincident with the given line, or on the side remote from J, may 
be termed superior or superjacent lines of route in respect of the given line. All lines 
are thus accounted for with the exception of those which cross the given line passing 
from the side towards J to the side remote from J, or vice versa ; these may be 
termed transverse lines in respect of the given line. 
Art. 42. I am concerned, at present, with those lines which are subjacent to a given 
line, though it will be remarked that the superjacent and transverse lines also 
suggest questions of interest. A given line of route defines a bipartite principal 
composition 
(mi Yh • • •)> 
and a unipartite south-easterly partition 
\P-Ih Y-V\-'¥z > • •)• 
The bipartite compositions and the unipartite partitions, defined by the subjacent 
lines of route, are termed subjacent to the given composition and the given partition 
respectively. 
We may draw a number of lines of route, each of which is subjacent to the given 
line and not transverse to any other of the number. We thus obtain what may be 
termed a subjacent succession of lines giving rise to a subjacent succession of 
unipartite partitions. 
These regularised partitions may be 
. . .), (b^bo . . .), ( c x c 2 c. 6 ...).... 
and they are such that the partitions 
i.. .), (a 2 b 2 c, 2 . . .), (a 3 & 3 c 3 .. .) . 
are also regularised. 
It is clear also that the subjacent succession of lines represents the multipartite 
partition 
^oqot^(z 3 . . ., b-yb 2 b ^..., cqct>c 3 ..., .....) 
of the multipartite numbers 
( a i + + c i H~ • • -j c h d~ b -2 + % + .. ., a 3 + 6 3 + c“ 3 + . .., . . .). 
This partition may be termed “ graphically regularised ” by reason of its origination 
