378 
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
The conclusion to be drawn is that 26945 has been wrongly allocated to S, (5) by 
Kayser and Kunge. It is further in favour of this that the difficulty as to the 
different limits for the S and D series referred to in [Part III. p. 404] is got rid of. 
The link 919*58 above comes in in the cycle from 30806 in which a number of doubtful 
links must occur. This one is clearly real, and it suggests that the adjoining one on 
the map for S (3), 963 is a false one. The lines 27140, 28103 would then come into 
the S (4) system. The map for S (4) has been drawn thus : 
AgP (1). P, (1) = 3047173, P,(l) = 2955P26, VP, = 30644*66, VP 2 = 3156510. 
The linkage stretching from P 2 (l) is by far the largest, over 280 lines appearing as 
linked together. In many regions the linkage is so complicated that it seems impossible 
to exhibit it in one map. Four maps are used to show the connections, a chain leaving 
a map at one line and passing on to the same line on another map. Amongst such a 
large number there must be many coincidences, and in fact the presence of some of 
them is shown by cycles in which the links do not identically annul one another. 
There are also a few cases of chains which appear to pass on to other independent 
linkages. It will be necessary, therefore, at first to make an attempt to clear up 
these ambiguities. 
The following cycles occur :— 
1. 37574-43720. In map (i.) there is a chain from 37574 (col. k) to 43720 (col. p) 
with links :— 
— b + c + c—u—v + e—b + e + c + v = — 2b + oc+2e—u. 
In map (ii.) the same lines (col. i to col. n) are connected by the chain 
—e+b+u+v+v—e+u = b — 2e + 2u + 3v 
If these are supposed all normal, their sum would be 6130'31 in (i.) and 6145*55 in 
(ii.). Now 43720'43 — 3757477 = 6145*66. Although not an absolute proof, this 
exact agreement with the total links in (ii.) is almost convincing that the pseudo 
link lies somewhere in the chain in (i.). But the links in (i.) are most of them good 
modified links. Thus going back from 43720 (see map (i.)) 2619'25 = v ( — d) + T9, 
96375 = c ( — 2d) — 79, 3776*44 = e(-2d)+*26, 92P01 = b + '54. The ’54 is partly 
due to error ’26 in the previous and a further —'28 on the line 37282, which makes 
all the neighbouring links better. 3775'55 is e( — 2d) — *68 or e( — d) +1*9. 1*96 is large, 
but if 33506 is 1*96 less (d\ = 77) all the other links converging on it become normal, 
viz., e, v. The link 2456*44 is n(d) + *05. The next link 964*80 is larger than the 
majority. It is c( — 3d) + '28. We might be doubtful about this were it not that the 
next is 960*08 = c (3d) — *48 and their sum 2c — *20 or two normal c links (example of 
series inequality). The last is 921*78 — b ( — 2d)+ '12. They all, therefore, apparently 
fit in well. The d displacement in the p links is, however, so small that little weight can 
be given to their agreement. Now a glance at the probability curves (Plate 4) shows 
that 921*75 and 964 occur with frequencies less even than the calculated chance 
