DR. W. M. HICKS:.A CRITICAL STUDY OF SPECTRAL SERIES. 
407 
but even should it be in error the doubt does not affect the argument as to the 
relative displacements of the different lines, as denotes the displacements relative 
to 19889. The final results are given in the following table :— 
1 
j 
19880 
2 
19942 
3 
19959 
4 
19989 
5 
20017 
6 
20021 
7 
20029 
8 
20041 
9 
20080 
10 
20107 
11 
20305 
12 
20312 
i 13 
20320 
14 
20333 
15 
20443 
16 
20454 
17 
20467 
18 
20470 i 
19 
20500 ' 
20 
20529 
21 
20559 
22 
20581 
23 
20596 
24 
20636 
j 
8OA2 + 2y8i - 5^ 
8OA2 - Si with y = 
8OA2 + 2?/Si - S - 
8OA2 with y = 
80A2 + 2.y8i+ IIS + 
„ ' + H8+ 
), + 2S — 
,, + 2^6 + 
), + 6 JS — 
„ +10S - 
,, + 15|S + 
„ +16S + 
„ +162^ + 
,, +17JS- 
81A2 + 2z/8i+ 4^S — 
11 + ' 5f S + 
„ + 6S - 
,, + 68 + 
„ + 72^ + 
+ 98 + 
+ 10^8 + 
+ 1118 + 
+ 12|8 + 
+ 1418 + 
66 + 14y + 
4 
61 + 16y + 
0 
77 + 17y- 
52 + 17?/- 
31 + 17y- 
50+ 17?/- 
2 + 18 ^ - 
33+ 18y- 
14 + 21y- 
82 + 21y- 
9 + 21?/- 
50 + 22y - 
11 + 24y- 
41 + 24y - 
12 + 24y - 
76 + 24y - 
71 + 26y- 
80 + 26y- 
71 + 26y- 
10 + 26?/- 
ll + 26y + 
36 + 26y- 
■ 2 ^ + 6 {q-])) 
‘05^+ ,, 
•05^+ ,, 
•05|+ ,, 
•05|+ ,, 
•07^+ „ 
•11^+ „ 
•17^+ „ 
•46|+ „ 
•47^+ „ 
•47^+ „ 
•50|+ „ 
■3o|+ „ 
•30|+ ,, 
•45^+ 
•45^+ „ 
•48|+ „ 
'48^+ ,, 
■m+ „ 
■5U+ „ 
’53^+ „ 
•62^+ „ 
-4, 11 
-5, - 4 
4, 11 
4, 3 
0 
-4, 9 
4,-8 
-3, 1 
6 
2, 3 
-3, - 1 
6, 0 
0,-2 
2, 3 
0, 14 
-1,-7 
-4,-2 
3,-8 
0, 9 
2, 6 
-5,-7 
0, -11 
-6,-2 
-2, - 7 
5, 8 
0, -12 
-6, - 3 
-3, 4 
3,-5 
-3,-7 
3,-4 
-3, 2 
3, 5 
-3,-7 
3, - 4 
0, 10 
0, 11 
-1, 13 
4, - 13 
Of the above (14, 18) must be set aside at once: (14) because it belongs to the 
triplet set linked by 2e to the parallel set F (l), and (I 8 ) because it is F 2 ( 2 ). It may 
also be noted that neither have the prevalent separation 1780 to lines of higher 
frequency. Of the numbers on the right of the list, those in thick type give the values 
of y which bring the outstanding differences to the corresponding number in ordinary 
type. These differences must be due to errors either in f or observation. Since £ is 
small, it is seen that they must be capable of annulment by the observation errors 
6 (g —p), and must, therefore, at the maximum be < 12 . The smallness of f can be 
seen from the following considerations which connect it with the corresponding £ 
(say i') foi’ the 1864 F series. The limit of the F is 30725‘30 + f' with mantissa 
889322 —30'74^'. It is a d (l) sequent. If 19989 is a D line with y = 0, its sequent 
is 51025‘29+-^' —19989'72 = 32035‘57 + The mantissa of this is 879853 —30’29^'+6g'. 
Both being d sequents must differ by a multiple of the oun. Their difference is 
9469+ 3U-29f-3074^-6g and 15^-^ = 9470. Hence 
30-29^-3074^' =1 + 6 ^, 
or, 
£^ l-015f'+-03 + -2g. 
Thus ^ '3. Since the 1864 limit is determined as the mean of F and F series 
its value is subject to a very small uncertainty and ^ will be a small fraction. 
VOL. CCXX.-A. 3 K 
