DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 



413 



The results are given in the annexed table, with notes up to m = 7. Numbers in 

 brackets refer to values obtained when none have 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. 



The true F 3 (o) is 30740'17-f where is small. The other limits will depend on 

 oun displacements from this. Estimated from 20312 the source of F 3 (oo) these 

 displacements, expressed as multiples of ^ are 2 if, 14^-, 13f, 9f, 123-. Their 

 values can, therefore, be calculated with exactness relatively to F 3 ( cc ). They are 



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 Fj. 



For the first order, m = 2, considerable displacements are to be expected. Only 

 normal lines for F 2 , 3 , 6 are observed. The set (5) 18998'40, (3) 19515'81, (7) 19676'25 

 give close normal separations 517'41, G77'85. Now the limit of l is 30493'! 1 and 

 the denominator of 18998 calculated from this is 3'08890G 134'3 or a mantissa 

 = 1088906-134'3^+ 24p = 99 (10998'14-1'35+ -Zip) = 99 A 3 . ^The normal /(2) 

 sequent is 89 A 2 . There is, therefore, a displacement of 10 A., in the sequence term. 

 Further the defect in the separation 677 '8 5 from the normal is 1'28 whilst a S l 

 displacement on the sequent produces I'll. The lines in question, therefore, are 

 F, (2) (10A 2 ), F 4 (2) (10A 2 ), F 6 (2) (10A S -^). 



To find a representative for F 5 we may test (l) 18 332 '41, 290'35, (3) 1862276 and 

 (l) 18018'31, 289-60, (6) 18307'91 in which the separations refer to that of F 3 , F 5 , 

 viz., 290'40. The mantissse 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 

 4A 2 2S = 70<S = 42770. An observation error of d\ = '03 would make this exact. The 

 lines in question may therefore be F 3 (2) (70S), F 5 (2) (70(5). So also it may be shown 

 that 18466'47 is F 2 (2) (-2A 2 ). 



With the F difference-series occur also the P 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. 



