DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 369 
them of the same value as the others, and the smallest displacement of ^ on VP(l) 
produces separations of far too high a value. It is necessary to look for some cause 
which will produce variations of from 2 to 10 in all cases. Such a cause would be 
small variations in N. The condition is also fulfilled if, instead of making the links 
depend on displacement operations by A on p, s, the operations are made on values of 
p, s, already displaced by small multiples of S l or S. For example, a' = — p(;r<i + A) 
+p(x$), b' = —p(xS) +p(xS— A), &c. The calculated changes are found to be as 
follows :— 
Ag (with xS). Au (with xS{). 
a’ — a 
- '61x 
- VI x 
b'-b 
— ' 61 x 
- 2'VJx 
c' — c 
— "66a* 
- 275a; 
d'-d 
- 7lx 
- 3’63x 
e! — e 
— 2'59x 
— 10'25x 
v! — u 
— 2'25x 
- 9T 7x 
v' — V 
-2%7x 
— 13'32x 
The x refer to multiples of S for Ag and of <b for Au. The reason for the difference 
of treatment is that where those of S x might occur in Ag the change would be much 
smaller than the observation errors, whilst if any of 8 occurred in Au the changes would 
be larger than those included in our purview. In constituting the formulte for the 
lines of certain linkages carried out below this method has been adopted, when the 
variations are too large to be ascribed to errors. This assumption, however, must 
not be considered to be adopted as the real explanation. This is left open. The use 
of the assumption enables the dependence of the lines on the p and s terms to be 
shown conveniently in tables. At the same time it exhibits clearly the presence of 
these small disturbances when existing, whatever their real cause may be. In many 
cases also after appearing in the tables for certain lines they disappear again for 
succeeding lines, and so have assisted in locating the latter. 
In the case of the long link e, which is the sum of the four short p links, it might 
be expected that some indication of the nature of this displacement might be given 
by considering how the e, or the enlarged e links depend on the displaced small links 
a, b, c, d. The number of the e links being large, and the number of possible 
arrangements of the small links within it being 4 !, the labour of testing them is so 
great that a complete discussion has not been attempted. The chief difficulty is that 
only a few of the small links in them are normal so that the others do not appear in 
the general lists of links, and the whole spectra have to be examined afresh. In the 
case of the AgS (3) and AgD (4) linkages no complete sets were found. In the 
AuP (l) of the links beginning at 25736, 20100, 37357, 41172, 26236, 43497 (see map 
AuP), complete sets were found for 26236 only. In the AuS (3) of those beginning 
at 25387, 20776 (Si 3), 23805, 24592 (S 2 3), 27783, 30109, 24477 (see map AuS (3)), 
3 F 
VOL. CCXVII.—A. 
