382 
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
are many of them very long, and naturally the presence of mere coincidences with link 
values is to be expected to a greater extent than in silver. The result is that in many 
cases a chain will appear to run from one linkage to another. No attempt, however, 
has been made to locate the spurious links, as such identifications would be even more 
uncertain than in silver. It will be sufficient to indicate where a chain runs on from 
one map linkage to another. 
AuS (3) . Sj (3) = 20776-50, S 2 (3) - 24592T7, s' = YS (3) = 8688’68. 
At the first glance the presence of a large number of meshes involving the long 
e links is noticeable. In the notes to the maps it will be advisable to follow the lists so 
far as the formulge are written down. The residual differences between the observed 
and calculated values might in many cases be reduced by admitting displacements, 
but it has been thought better in the present state of knowledge to seek for simplicity 
}f a first approximation than exactness where the observation errors are quite unknown. 
The separation of S 1; S 2 is practically exact. There seem to be several links about 
3188, which have been introduced on the evidence of meshes. From S 2 (3) 3199 is 
regarded as 3198‘54 = «( — 2§ 1 ) and the next a link as 3188'34 = a(S), but the two 
together can be regarded as 2a (S x ) in the value for 22813 and succeeding lines. Also 
the 3199 and the 3191 from S 2 (3) form a good example of series inequality, their sum 
being practically an exact 2a. The successive links 11347 and 14931 No. 7 would not 
have been admitted singly, but the excess of the first and equal deficit of the second 
point to a displacement in 28525 which does not recur in the preceding and succeeding 
lines. It may be noticed that the order of operations is S x to S 2 by a displacement — A 
in the limit, then a modified —a link, then by another — A displacement, again by a 
modified —a link. In the list as originally drawn there was no further — A displace¬ 
ment which should be 5635 ahead. But there is a line 28439 which is 5626'92 ahead. 
This is just as much in deficit from a d link as the previous 3188 from a. It is there¬ 
fore a d link with 3188 made normal, and clearly comes naturally into the system. 
This again has a modified —a link. If the law continued an extra — A displacement 
in the limit would give a line 6993 ahead, but none is found. It would correspond 
to a p( — 4 A) term. The sequence is also continued back to 23966 by a modified a 
and thence by A displacement to 20768. Thus from 20768 we have the sequence— 
20768 - p (A) + X (say), 
23966 = p + X, 
20776 = p-a {2S 1 ) + X, 
24592 = p(-A)-a{2S 1 ) + X, 
21392 = p( — A) — a (2<^)—a ( —2ci 1 ) + X = p (-A)-2a + X, 
26000 = p ( —2A) —2a + X, 
22813 = p(-2A)-2a-a(S) + X, 
28439 — p ( —3A) —3a + X, 
25249 = p ( —3A) —3« —a (<i) + X = p (-3A)-4a (^) + X. 
