362 
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
suggested by the case of Cu in which their values are comparatively small. But it 
required the support of the larger values afforded by Ag and Au to firmly establish 
the relations. Similar relations have been found in many other elements. The spectra 
of the rare gases from Ne to RaEm are built on a precisely similar plan, and in fact a 
map for some of the Kr lines was drawn many years ago on the plan of those given 
here for Ag and Au, although at that time the origin of the differences was not 
known. 
The notation used is that of previous papers in this series. The letters p (m), s ( m) 
are used to denote the sequences N/(m+ f ) 2 which in the alkalies give the variable 
parts of the Principal and Sharp series. The wave-numbers of the Principal series 
are then s (l)—p (m) and of the Sharp p (l) — s (m). The doublet separations are 
due to denominator differences denoted by A, say m + f and m + f — A. The 
quantity A is a multiple of the oun, = 90‘47W 2 , where 100W is the atomic 
weight. As however the quantity <1 = occurs very frequently, this is generally 
used. When xS is added to the denominator in either of these sequences a new line is 
produced, and it is said to be laterally displaced. Thus, P (m) denoting any line, if 
xS is added to the variable part (the term in m), the new line is written P (m) ( xS), 
but if added to the constant term (the limit) it is written (x§) P (m). Regarded 
from this point of view the two sets of a doublet series are regarded the one as 
a displacement of the other. Thus P, (m) — Pj (m) ( — A) ; S 2 (m) = ( — A) S x (m) ; 
D 12 (m) = D n (m) { — xS ); D 22 (m) = (— A) D n (m) (—xS).* 
In the following, for the sake of shortness, when p, s are used alone they stand for 
the limits jo(l), s(l). Line separations are always given in thick type, and, in 
general, the decimals of wave-numbers are omitted. 
It will be seen later that the structure of the spectra depends on long series of 
constant differences in wave-numbers (or frequencies). As the total number of lines 
in a particular spectrum is very great there will always be a certain number of chance 
agreements for any given difference. In forming an opinion as to the reality of the 
connection indicated by a given set of such differences it will then be necessary to 
obtain an estimate of the number of such coincidences to be expected, on the 
supposition that the lines form a purely chance distribution. Owing to observation 
errors exact differences will never occur. Suppose that the difference being investigated 
lies between n ± x, where x is a small quantity, and suppose that the spectrum under 
consideration contains p lines whose wave-numbers lie between Nj and N 2 , N x < N 2 . 
The average separation between successive lines will be (N 2 —N ; )/(p— l). The 
chance of one falling in the space 2x is therefore 2x (p— l)/(N 2 —Nj). Taking 
each number in succession from the smallest, this is the chance of finding a line n ± x 
ahead of it. But clearly we must stop when we reach a value N 2 — n. The probable 
number of lines contained in this region is n (p— l)/(N 2 — Nj) and the probable total 
* There is reason to believe, however, that the series P 2 ( m ) and the satellites are the normal lines, and 
that Pj (to) should be written P 2 (to) (A); D n (m) = D i2 (to) (xB), &c. 
