DR. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 
S97 
Ill (4) the separation is 1767-09 - . 1 -10-81. If it corresponds to a real the limit 
will be ( 5 ^) D ( 00 ) which reduces by 10-7 and makes the limit about 212 le.ss. The 
mantissa would be 82 A 2 +7|-(1 +71. This difterence ( 7 l) from an exact multiple of di 
shows this to be impossible. 
The discussion of the 1864 series has definitely shown it to be of the F type, and 
has given the limit within very small errors. This limit is one of the d (l) sequents 
of the diffuse series. Its mantissa was found to be 81A2-2|-(5 or 80 A 2 + 15 |-A It 
gi’ves one fiim staitmg point for the discovery of the D series. The results just 
obtained indicate the lines which through their dependence on multiples of A., give 
the origin term of the diffuse sequence. They point, as we have seen, to the existence 
of several di.splaced, or parallel, sets of diffuse series. It is possible to show definitely 
that these exist, even if there be some uncertainty as to the lines occupying the 
position of ( 1 ). In the normal case with a single diffuse series, the ( 1 ) is always 
the sti digest line of the series. Also 111 the normal type the D limit is the same as that of 
—here 51025-29 + f. When however displacement occurs, the energy of a single line is 
dispel sed amongst several others, and a line corresponding to the normal may be weak, or 
even too faint to have been observed. As a matter of experience also it is found that the 
lines of low order (m = 1 , 2 ...) are subject to these displacements in a much greater 
degree than those for higher orders of m. Now there are a number of lines, which by 
their position and absence of j/j separations to stronger lines have the appearance of 
being (l) lines. If they are, their mantissse must differ from multiples of A 2 (in 
the present case 8 OA 2 ) by multiples of the oun. The fact that they may do so does not 
of course prove that they are lines. If they do not do so it proves that they are 
not. I hey may however in the latter case belong to a displaced seiies, satisfying the 
multiple law when the proper displaced limit {ySi) S ( 00 ) is employed. This gives us a 
method of testing as to what displacement a given line may correspond. If our 
calculus were already fulty established the next step would be to apply this test to 
the above lines. But in reality we are testing our calculus to see if it can l)e firmly 
established, and our immediate aim must be to obtain independent evidence for the 
existence of parallel series. For this immediate purpose it will only be necessary to 
apply the test to two lines, the general question being postponed for the present. 
In the first attempt at arranging the Du series the strong lines ( 8 ) 20559-08 and 
( 10 ) 38366-36 were taken for m = 1 , 2, and the formula calculated with the limit 
D ( 00 ) == S ( CO). As will be seen immediately, this gave satisfactory agreement with 
sounded observed lines up to wi = 15 ; and as a matter of fact this series was used to 
test for the parallel sets displaced (±2^i) S ( co) on either side of it. Now the formula 
constants for a set only vary slightly if the wave-number of the line chosen for oyi = 1 
IS changed considerably. This therefore did not prove definitely that 20559 is the 
correct Du (l), and as a fact it does not satisfy the multiple test. Its mantissa is 
897337 — 31 14f That of the hue 19989, which is shown above to be the origin of the 
normal D .set is 879853-30-29^. The difference is 17484 = 28i<5 + 7 l. This is as far 
