DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
371 
is an arc line. Apparently, therefore, there is only good evidence for all the inter¬ 
mediate links in Nos. 1, 3, 5, 6, 8 and 9. In applying the displacement rule to the 
e links in the formulae lists given below the numerical values allowed are supposed 
to be e = 1725378 and e(—^) = 17264'03—in other words, e is taken as an inde¬ 
pendent link and not as the composite a-\-b + c + d. If it is composite each of the 
a, b, c, d may be modified, and values of e intermediate to 17253 and 17264 may 
enter. If then the wave-numbers were known accurately to unity (dX = '5 for long 
waves to ‘05 for short) these ought to serve as tests of the kind of displacement 
supposed. Take, for instance, (3) as an example. It is seen that e = 17262'04 
= e( — dj) —2, and the small separations are a(3^) —'54, b ( — <^) — '39, d( — 3(^) + l’59, 
c — '26. Thus if e is really composite it should equal a (3d x ) + b (— d x ) + d (— 3dj) + c, and 
its value should be '40 larger— -i.e., it is already correct within very small error limits. 
If, however, it is independent, the link 5647, which is just as far as it can be from 
d ( — 3d x ) or d ( — 4(1!), would be spurious, 23966 and 27783 would be linked to 20776 and 
33431 to 38038, and the error —2 in the long link would be divided between 
observation errors on 20776 and 38038, say 20776 -'43 and 38038 + l'57, giving 
respectively dX = '10 and —'ll. All this argument, however, goes on the assumption 
that only one displacement can take place at each step. Thus in passing from 27783 
to 33431 the existing a (3dj) may displace to a (— dj) and the d link be d (— dj), producing 
a line '90 (dX — '08) less than that observed. The next link would be c( — S 1 ) when 
38038 is corrected by 2 and then the transformation to e (— d x ) is completed. That these 
double displacements take place is not unlikely since the successive lines in our lists 
may be quite independent of the neighbouring lines, the chain links merely serving 
as a sort of Ariadne’s thread to discover them. 
A corresponding treatment for all the sets gives :— 
dX. 
3. e(-di)-2, a(3d 1 )-'54, 6(-d 1 )-'39, d (-3^) + D59, c-'26 . . 14 
5. e (— dj) —' 9, a(2d 1 )- , 66, d (-3d0 +159, c-'26, 6(-d,)-' 93 . . '05 
6. e(-d 1 )-3'86, d ( —3d : ) +1'59, c-'26, 6(-d 1 )-'93, a(d 1 )-*27. . 19 
'8, e + 2'48, 6 (<&!)+'61, c+'31, d( —d 1 )+15, a-K95.-'20 
9 (a). e( —dj) —2’54, 6 + 11, c( —d 1 ) + ‘09, cZ( —d x ) —1’83, a( —d x ) —17. . . 12 
9(6). e + 3'54, 6-'94, c + '69, d + 1'82, a+1'53.. -17 
In the above the value of dX is the correction to be applied to the last line to make 
the long link e or e( — S 1 ) exact supposing the whole error thrown on that line alone. 
In 9a, 9b only are there indications that the small links show displacements the same 
as those given to e, but this cannot be the case for both sets as the two last links 
must be the same. The discussion of the cause of the enlarged links must therefore 
be left open until more accurate measurements are attainable. 
There are a large number of more or less parallel linkages which unfortunately it 
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