468 



Prof. W. M. Hicks on 



The values of the / sequence as found above are here 

 collected, together with the values of their denominators. 



/. 



27385-67 



= 1 \ 27349-31 



[ 27445-36 



12257-10 



2249-48 



m- 



J 122 

 1122 



6880-35 

 6877-20 



2-001208 

 2'002537 

 1-999030 



2-991400 

 2-992313 



3-992533 

 3993446 



m = 4, 

 m=5, 

 on=6, 

 m=7, 



4402-00 

 3058-3 

 2244-33 

 1715-9 



4-991473 

 5-988440 

 6-992044 



7-994780 



The real F series depends on limits d 1 (l), d 2 (l), and 

 if the / sequence is the same as in the above we ought 

 to be able to find them. There are several possible values of 

 the d(l) as we have already seen, the most stable apparently 

 being that based on 132 A. It gives d 2 (l) = 28417* 66 + a?, 

 x being a small uncertainty due to inexact N. The F(l) lines 

 will be in the neighbourhood of ?i = 1000, and beyond the 

 observed region. There is a line at n = (3) 55800*5 — 30d\, 

 about the position to be expected for a F (1) line, and as it 

 occurs by itself we should expect it to be F 11 = ^ 1 (l)+y 1 (l). 

 But we have seen the most probable satellite separations are 

 about 25 in both sequences, but opposite, so that d 1 = d 2 — 25 

 and f 1 —j 2 -{-25. The lines d l +f 1 and d 2 +f 2 would there- 

 fore be almost coincident. Hence /2(1) = 55800*5 — 30<^X 

 -28417*66-^ = 27383-8-30d\-.z','which recalls the value 

 obtained from the second <^(2) + /'lines adduced under m = l 

 above. We shall see reason below to put F 2 (c© j = d 9 (l) 

 = 28410-26 or ^=-7-4 about. This makes / 2 (1) =27391 

 — 30d\. In the table above the mantissa of 27385*67 is 

 given as 1001208, and this is 137x7308*919, very close 

 therefore to 137 A. It suggests therefore that 55800 is 

 N/(l-f 132A) 2 + N/(1 + 137A) 2 and thus the normal F 2 (l) 

 line. Its discussion is deferred for fuller treatment of the 

 general theory. 



m=2. With ^(1) = 28410*26 and satellite displacement 6 8 

 dj(l) or Fi( x> ) is 25*37 less, - 28384*89 + . With 16126*12 

 as F n this gives /i = 12258*77. F 22 gives f 2 = 12245*10. 



m — 3. This set is important as with the known /i(3) 

 = 6880*34 and 72(3) =6877-16 the limits F^qo) and F 2 (co) 

 come out to 28382*96 and 28410*04. The latter should be 

 correct within a few decimals, but the F x (oo ) depends on a 

 doubtful spark line (X = 4177*87). 



m = 4. The whole region for F is clear of lines except 

 (4/i)23929*06. This is 51*90 behind the calculated. If it 



