394 
DE. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
differences of the inantissse, which may themselves be affected with errors ± 1 due to 
using 7-ffgure logs. 
The differences may be represented as follows :— 
5 (10996-8+ 5-6(7i^i--52^+•2)-(5i = 
2 (l0999-0--5c?i/i+13-5d»^2±-5) + 6<li = 2A2+6^i, 
where to a first approximation we know already Agis 10998±1. The sum 
= 7 {l0997-4--53^±'l + 4(dz/j + r7i.2)} + 5^i. 
Quite small changes in dv^, dv 2 can therefore make the connections with A 2 exact, 
and since the imdtiples 5, 2, 7 are so small any small errors in the approximate value 
of Ao can have no effect. If we use the as found from the occurrency curve 1864-5 
dv^ — - 4 , the number in the first bracket is 10999. 
The agreement with both is so close to the relation indicated that it speaks 
strongly in support of the D origin of the limit, and the outstanding small differences 
may Ije left for the present. 
If now the other supposition be tested, viz., that the separations arise from a S ( 00 ) 
source = 51025-29 + ^, the three denominators are found to be 
1-466091-14-36^ 
1-440024-13-61 [i+dv^) 
1-428884-13-30 
26067--75^+13-61c?i/i 
1116 0 --31- 3 1 +13-30(7i.2 
and in the 20312 set the S(co)must enter as a negative quantity since the 
separations are there in inverse order. In this case ^ may be considerable. The 
differences may be expressed as follows :— 
2A. + 6f^- 53 - - 7 5^+ 13-61 dri; A 2 + (^1 + 9 - - 3 1 13-3di/2- 
No permissil)le values of v^, v., can make these both multiples of the oun. If it is 
possible to do so by a proper choice of ^ the latter must satisfy 53 +-75^= 1537?i, 
— 9 +- 31 ^ = 153w, where m, n are integers and 153 is the value of the oun. This 
requires 22 = 47-4wi—115n. A suitable solution is m = —2, n = — 1, which requires 
f = 480, or, say, series limit = (ll^Q S (co). This method of explanation looks then 
improbalffe especially when taken with the more natural one above. It may be 
concluded with some confidence that the series in question is of the F type depending 
on D series for limits as in the usual way. 
In the suggested lines for m = 1, found by sounding with 2e the mantissa of/(l) 
was found to be 90 {10998-8 —- 4 ^+-4d?i.}—^. ^ is small and the term involving it 
may be omitted. The error dn in 3010 may, however, amount to a few units because 
the lines on which it was based were assumed to depend on the limit 30725, whereas 
there is the possibility that they might belong to one of the parallel series found in 
