DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 365 
The exact values of these links depend only slightly on the exactness of the values 
of the limits, for if the latter change by £ A also changes with it in such a way that 
the consequent changes in the links are only small. For instance even with the large 
values of Au, the links a...e become a + 0, 6 + 0, c— '03£ d — '07£, e—'08£. The 
links u, v, however, depending on a limit which does not affect A, show larger, 
although still small, variations, viz., u of — '24^ and v of —'8f. 
In the case of Ag, the greatest and least values of the wave-numbers tested are 
N 2 = 50611, N x = 15691, whilst the total number of lines is p = 600. Substituting 
these Values in the formula given above the number of probable coincidences with a 
given link n ± x is 
20x (1 — —-—-) 
V 35000/ 
The corresponding values for Au are N 2 = 53697, Nj = 15923, p — 750, whence the 
probable number of coincidences is 
In order to form a judgment on the question whether the observed coincidences are 
due to a real effect or to mere chance, the series of curves Ag 1-7 (Plate 4) and 
Au 8-14 (Plate 5) have been drawn. The actual observed differences are represented 
as abscissse, a dot being placed for each case. For any particular difference, the 
number of dots within x on either side of it are counted and entered as an ordinate. 
Joining the tops, we get a broken curve which represents the frequency of occurrence 
of the various values of an observed link. A dotted horizontal straight line drawn 
at a distance above the line of abscissse, calculated from the above formula, gives the 
probable frequency on the supposition that the values are due to pure chance. The 
curves are drawn for the two cases of x = '4 and 1. The curves for the long e links 
are made to cover a wider region than the others for a reason considered below. 
The v link for Au has also been extended further in order to see how the observed 
differences behave when they are away from the immediate neighbourhood of the 
true link. As is seen, their frequency is much less than the calculated chance, in 
agreement with the reasoning given above that the chance distribution must be less 
than that given by the formula. In the Ag set the curves show that the four small 
links, a , 6, c, d are clearly systematic, with numbers more than twice the maximum 
probable on chance and practically agreeing in value with the calculated results. 
The three larger links also show the same effect with the distinction that they appear 
to be arranged, about three maxima, roughly two units apart for u, v, and three units 
for e, with e = 3771 itself actually at a minimum. In Au, the existence of peaks is 
even more marked. They are probably all real effects and indicate that certain small 
displacements will often simultaneously occur with the change of link. 
The existence of observation errors will flatten out any peaks if they exist in the 
9'8cc 1 - 
n 
38000 
