388 
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
doubt. The formulae for this chain have not been written down, partly because 
of the prevalence of the small modified u. v links, and also because a few false cycles 
show that some spurious links occur. In the X. i. map we have an example of a true 
cycle from 23354 (c. 11), viz., a + d—a + c—d—c — 0. 
AuD (2). AuD n (2) = 17125-54, AuD 21 (2) = 20858'97, VDj = 12339‘64, 
VD 2 = 1242175. The linkages appear amongst the X maps. 
The line D n (2) appears in the X. ii. map in col. a. From this a chain proceeds to 
the top of the map consisting, like other D linkages, largely of p links. They fit in 
well and the formulae are simple. Another chain extends downwards into lines which 
at first sight appear analogous to those in X. i. There is, however, a remarkable 
long cycle which rather seems to indicate that the majority of the lines in X. ii. belong 
to the D (2) series. The line 28620 appears at the top of the map at (e. 2) and also at 
the bottom, shown in the inset map at (i. 4). A true cycle runs between. Starting 
from (i. 4), and supposing all the links normal, they run in the following order to (e. 2) 
d + u— a + c — b— u — d + a+v—e + v — c + d— v — a — c + b + a— d + c + e—v in which each 
kind of link is annulled. Although not an absolute proof it produces conviction that 
all the links are real. The whole series forms, so to say, the backbone of the whole 
map in which only the side out-liers may be doubtful. Assuming this it is easy to 
suggest the position of certain spurious links. For instance :— 
(a) The chain from the true X 21 line (15923) joins this cycle at the line 42016 
(f. 7) by the link 3197‘66. This joins the corner of a mesh which is 
undoubtedly an X 21 set. This link is probably spurious and is marked 
doubtful. It differs as much as possible from a displaced a link. 
(b) From 26303 (d. 3) a connection passes to the P map. It is the case already 
referred to in the discussion of that linkage. The connection must be 
spurious, and the false link is probably the 11332 on which doubt was there 
cast. 
(c) A long cycle also stretches from 24273 (g. 2) to (i. 8), but this is a false one. 
It only has one link in common with the other, viz., that connecting 42397 
and 47004—( g . 13) to (h. 9). 
D 21 (2) appears in AuX. iii. b. From it a chain runs down to 38457 which has been 
taken to be on an X chain. The links, however, all seem good from the present point 
of view. If the I) 21 (2) and X lines are correctly allocated there must be a spurious 
link here. If the links are real then D 21 (2) and X are not independent, and one at least 
has been incorrectly allocated. It is at least interesting to notice that the formula for 
38457 depending on D 21 (2) involves no displacements and reproduces the observed 
value by d\ = —'05. From 26494 another chain is drawn upwards. The formulae 
are written down for a chain up to a terminal at 26566 from which side lines can 
easily be written down. The two long e links are naturally represented by e( — <b) 
with simultaneous displacements in other links. At 41500 (f. 3) there is a typical 
