The following notation is used throughout this section and the next:
where 
is the number of XMM sources in the sample under
consideration.
where 
is the number of candidates for the
XMM source.
to the position error
of the 
XMM source in either direction.
When an RMS position error is defined for the whole sample it is referred to
as 
.
is the normalised distance between the 
XMM source and
the 
candidate for that source in arcsecs, i.e. 
The actual distance 
is often taken
as the offsets along 2 perpendicular axes added in quadrature, although in
our calculations rigorous great circle distances are computed.
is the search radius, i.e. the maximum value of 
for
an object to be considered as a candidate for that source.
represents the probability of event A given event B.
Hence 
is the probability that the 
candidate, found
at distance 
from the 
XMM source, is a true identification
for this source.
Similarly 
is the probability that the same candidate is a
chance candidate, i.e. not the true identification.
On the other hand, 
is the probability that a true
identification exists at distance 
from the 
XMM source.
Note that in the probabilities mentioned here, in the cases where magnitude
as well as separation is taken into account, we should use (for example)
(where 
is the magnitude of the 
candidate) rather than 
.
.
However in some cases it is desirable to take into account the magnitude of
the optical or infrared candidate as a bright X--ray source is likely to have
a bright counterpart. Thus we define 
as the background density of
optical or infrared objects with magnitude less than or equal to M. When
considering a candidate labeled j, 
is used.
.
.