DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
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probability. The spurious link is therefore probably one of these. But if the 92175 
link is retained there is a sequence through 37574 of e and h links e + e — b + e—b + e, 
the regularity of which would seem to have a reason, and would include 921 as real. 
Provisionally, therefore, 960'08 or 964'80 (or both) is taken as spurious. 
2. 38577 in (ii. j) to same line in (i. 1). There are two cycles in this case. In one 
the complete cycle of links is, starting from (ii.), d + v—c + e—v + u—e + u — c—v + e 
(43720 (ii. n) pass to (i. p)) —v — c — e + b—e f v+u = b — Sc + d—e + Su—v. In this the 
second c link is 96671. It is so excessive that it should scarcely have been included. 
We shall probably be right in rejecting it. The other is, starting from (ii.), d + v (pass 
to(iii. m )) +c — e + c—u — u— 6(passto(i.))--d+v + w = — b + 2c—e—u + 2v. Provisionally, 
the b link 922'51 may be marked as spurious. 
3. The cancelling of the link 966 in the preceding case also annuls two other cycles, 
one a long one from 35435 in (ii. g) to 42203, thence to (iii.), and the other from 35435 
through 33506. 
4. 29737 appears both in (i. e) and (iii. c). The cycle is annulled by omitting the 
large u link 246270 in (ii.) after 35150. The linkage connected, however, with 29737 
probably belongs to D (4) to be considered immediately. 
5. 29084 on (ii.) is connected across to 31703 by a link v = 2619, the latter line 
belonging to 29737 just considered. The cycle is annulled by omitting the abnormal 
link 958'25 after 35435. 
6. There is also the case given as an example on (p. 373) and there considered. 
With the links indicated above left out, the P linkage is now singly connected, 
except in the case of true cycles. There remains, however, one chain which leads on 
to the D (4) linkage. The line 29084, appearing in the cycle (5) just considered, is 
itself a part of the D (4) system. It is linked to D 12 (4) and D n (4) by a mesh, and 
it is practically certain that this connection is real. In this case a certain set of lines 
attributed to the P linkage must really belong to D and a false link must occur. 
29084 is joined to 31703 by 2619'80 — v ( — <5) + '74. It is impossible to notice the 
sequences of this link with the p links on the left hand top corner of (iii.) without being 
convinced that they form a connected cluster. It is difficult, however, to see where the 
false link comes anywhere on the chain up to 42203 which is connected in several ways 
with P lines, or at least with lines attributed to P. Further discussion is deferred 
until the D (4) linkage is considered. 
With the lines in the map now singly connected it will be possible to write down 
their formulae in terms of the p and s values. As, however, in such a large number of 
lines there must still remain a number of spurious links, and as the existence ol one ol 
these will invalidate all the formulae for lines occurring later, only the first sets 
starting from P L and P 2 are given here. From Pi a single chain stretches which is 
remarkable for the long series of enlarged e links. They are all close to their 
mean 3778'44, which only differs from e ( — 3$) by '33, and at the same time the link 
to P x is u ($) + '25. This chain, therefore, appears to support the theory ol link 
3 G 2 
