398 DE. W. M. HICKS : A CRITICAL STUDY OF SPECTEAL SEEIES. 



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 

 D n (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 S } on the limit changes its value by 

 10'62, so that (ySi)~D(<x>) = 51025'29 + ^-10'26t/. 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 80A 2 = 879856 + 80x. Also 3307 = ^ + 257. Hence the mantissa of 



20559 is 



80A 2 -80a;+ 17481 



in which x is small (about '2). Also from the consideration of 19989 above 



.= 0. 



Eliminating x 



= M(l53) 



in which p, p'<l and cannot exceed about 2. 



It is not possible to satisfy this with y = 0, 1, or 2. 

 With 



=-S -5 + 6 (p- p '-'tf = 0. 



If, then, 20559 be a D H line it belongs to one of (3^)D(c) ; and is definitely 

 excluded as a possible normal D u . 



The next line of higher frequency is the weak line (l) 2058 1'64. Its mantissa 

 is 898040-31-17^= 81A 3 +llf <J + 7-81a; + 6-5jt>-3ri7 = 81A 2 -ll|<5 within error 

 limits. This therefore passes the D n test. If it is the actual D H its weak intensity 

 is due to the numerous displacements for m = 1. If it be taken asD u (l) with 

 the previously mentioned (10) 38366'36, and the limit D(<), the series formula is 

 found to be 



n = 51025-29-N/jm--988854- 090 ?- 1 i} 3 . 

 / ( m \ 



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 



