DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
413 
The results are given in the annexed table, with notes up to w = 7 . Numbers in 
brackets refer to values obtained—when none liave been directly observed—by 
connection with linkages or displacement. It may be recalled that a link from an 
observed line to an expected but unseen one leads to the inference that the 
unobserved really exists, whereas a displaced line, when the displacement is on the 
limit, gives evidence only for the value of the sequent f{m), and when the displace¬ 
ment is on the/-sequent, is evidence to that effect alone. 
Ihe true F 3 ( oo) is 30740‘17 + ^ where ^ is small. The other limits will depend on 
oun displacements from this. Estimated from 20312—the source of ^" 3 ( 00 )—these 
displacements, expressed as multiples of <^1 are — 2l|-, —14|-, —13|-, 9 ^, 12|-. Their 
values can, therefore, be calculated with exactness relatively to F3 ( 00 They are 
Fi. 
F 2 . 
F 3 . 
F 4 . 
F 5 . 
Fe. 
30493-11 + 1-0121 
4-910 
59- 
30552-13+ 1-01^ 
4-927 
02 247 
30740-17 + ^ 
4-964 
•06 517 
31010-41+ -99^ 
5 • 030 
•30 537 
31030-58+ -981 
5-043 
•47 679 
31172-24+ -98^ 
5-090 
•13 
The numbers below the limits give respectively the changes produced in them by 
the displacement of one oun. The numbers in the last line give the calculated 
accurate separations of the corresponding F lines from F^. 
For the first order, m = 2 , considerable displacements are to be expected. Only 
normal lines for F 2 , 3 , e are observed. The set (5) 18998-40, ( 3 ) 19515-81, (7) 19676-25 
give close normal separations 517-41, 677-85. Now the limit of F^ is 30493-11 and 
the denominator of 18998 calculated from this is 3-088906-134-3^, or a mantissa 
= 1088906-134-3^+24^ = 99 (10998-14-1-35^+-24jp) = 99 A 2 . The normal /{2) 
sequent is 89 A 2 . There is, therefore, a displacement of lOA^ in the sequence term. 
Further the defect in the separation 677-85 from the normal is 1-28 whilst a 
displacement on the sequent produces 1 - 11 . The lines in question, therefore, are 
F/2)(10A2), F,(2)(10A2), Fe{2)(lOA2-<^i). 
To find a representative for F 5 we may test (l) 18332-41, 290'35, (3) 18622-76 and 
(1) I8OI8-3I, 289‘60, (6) I8307-91 in which the separations refer to that of Fg, F5, 
viz., 290-40. The mantissee difference for the first set is 5740, and the nearest oun 
multiple 9|-(^ = 5804 is outside error limits. That of the second is 42758 and 
4A2—2^ = 70S = 42770. An observation error of dX = -03 would make this exact. The 
lines in question may therefore be Fg (2) {70S), F5(2) (70(!). So also it may be shown 
that 18466-47 is F2(2) (-2A2). 
With the F difference-series occur also the F summation type. As their existence 
is a new fact of great importance the evidence available up to m — 10 is given. The 
results are embodied with those of F in the table. 
