398 
DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
as it can be from being a true multiple, and it is quite impossible to explain this by 
any observation errors in the two lines. For instance our maximum admitted error 
can only change the mantissa by 6 . If the test is valid therefore 20559 cannot be 
Dll (l)- Yet it has all the appearance of such a line. Is it a displaced one ? Suppose 
it corresponds to yS^. Now a displacement of on the limit changes its value by 
I0‘62, so that (//( 5 i)D(oo) = 51025'29+ ^—10’26^. If p denote the ratio of actual 
observation error to the maximum permissible, i.e., O = d\ = '05p, the mantissa of 
20559 with the new limit is 897337 + 3307?/+6p —31‘14^. The denominator of 
19989 is 8 OA 2 = 879856 + 80a;. Also 330’7 = ^^ + 25’7. Hence the mantissa of 
20559 is 
80A2-80a;+17481 +2^^1 + 257?/ +6 j9-31-14^ 
= 8 IA 2 +10^(5+2^^i + 68-81r» +257?/ +6j9-3ri4f 
in which x is small (about ±' 2 ). Also from the consideration of 19989 above 
-3-2-80a; + 6p'-30-29^= 0. 
Eliminating x 
n + 25-7y + 6{p-p')--A7i= M(l53) 
in which p, p'<.l and i cannot exceed about 2 . 
It is not possible to satisfy this with y = 0 , + 1 , or + 2 . 
With 
y = S —6 + 6{p—p') — ‘A7^=0, 
?/= —3 —5 + 6 {^—jp') —'47^ = 0. 
If, then, 20559 be a Du line it belongs to one of (+ 3 (^i) D ( oo and is definitely 
excluded as a possible normal Du. 
The next line of higher frequency is the weak line (l) 2058r64. Its mantissa 
is 898040-31-17^= 81A2+llf(l+7-81x + 6-5_p-3117^= 8 IA 2 -IIP within error 
limits. This therefore passes the D^ test. If it is the actual D^ its weak intensity 
is due to the numerous displacements for m = 1 . If it be taken asDn(l)with 
the previously mentioned (lO) 38366-36, and the limit D(co), the series formula is 
found to be 
n= 51025-29-N^jm--988854- 
The lines after m — 2 lie in the violet outside the observed region. To test them 
therefore recourse must be had to sounding, only the e.u.v. links have been used for 
this purpose. The results are given in the middle column of the subjoined table and 
exhibited in diagram (Plate 3). Details are given in the notes following the table. 
Lines were calculated down to m =15 and tested. The result may be regarded as 
conclusive in establishing the series, as well as increasing confidence in the method of 
sounding—a confidence which reposes not on a single coincidence, but on the 
recurrence of a large number of successive ones. As will be seen the agreement 
