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DR, W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
If, however, 18744 is really in the S (2) linkage, the second of the above would only be 
a coincidence, and 5740*86 would be a sum of links — p (A) —p ( — A) + 2p ( — 3A) of 
which no other example is known. The existence of the two combinations together 
forms a strong argument that 18744 is the D n line, and that the linkage relation 
is a coincidence. 
AgS (3). Sj (3) = 21413-37, S 2 (3) = 22333-79, s' - VS (3) = 9231‘23. In the map 
of this linkage there must be at least one spurious link shown by a complete false 
cycle, and two possible ones shown by two chains, one leading to D (5) and the-other 
to P(l). 
The links from the S (3) mesh back to 17874 (No. 6) are all good except the two 
successive 2613*74 and 882"58. They are admitted, however, because they are an 
example of series inequality deviating by the same amount (17) in opposite directions 
from exact normality. They clearly point to a displaced u (d) link for the first with 
the displacement annulled in the second. With the long e links stretching from 
S (3) to 28968 we get the enlarged values which seem frequently to be associated with 
the chief lines. 
From 28968 two chains diverge, one through 31429 ultimately leading to D(5) 
through 30393. There are no clear indications to show where the false link occurs. 
The first link u = 2461 "12 is so large that at first sight it might seem to be spurious ; 
but it falls in well with the others as may be seen from the formula list. It would 
look as if the series of long links should be 3773"59, 377877, 377877, i.e., u(—S), 
u (— 3d), u (— 3d), but that the last for some reason took u (— 2d) at first but completed 
the extra (— d) in the next link. The formulae in this chain, as will be seen, involve 
very few displaced links beyond those it started with, due to the above indubitably 
enlarged e links. In fact such only enter in No. 21 and in 25, 26, which latter form 
part of a mesh. On the other hand, as will be seen below, the formulae based on 
D(5) show very good links up to 29514 and even beyond. Provisionally I have 
placed the spurious link as that between 29514 and 30394 with a — 87872 in place 
of 880-77. 
lieturning to 28968 and considering the other branch chain from it, the second link 
918*65 is small, but is naturally accounted for by the long e—which entered as due to 
3779 after Sj (3)—changing to the same value as the second e link 3776 which 
entered immediately after it. The links 2455*51 and — 2461*79 appear at first sight 
too abnormal, they are exactlv explained by a return of the two long links after 
S,(3). which became equal in No. 27 to the value of that before Sj (3). From 
30806 starts the false cycle referred to above. In order to keep the map as open 
as possible the line appears in two places in it—here and at the top of the same 
column. In the absence of sufficient knowledge of the laws governing links it seems 
impossible to point out where the pseudo link occurs. The long links 3774, 3778 
which usually appear successively here appear separated by — b + c, although a set b + c 
appear as if a part of a missing 3778. This consideration would therefore favour 
