DR. W. M. HICKS: A CRITICAL STl'hY OF SPECTRAL SERIFS. 337 



going further it is desirable here to consider tin- nature of the cumulative 

 effects produced by errors in the values of 5, or of the limits, in the course of a 

 succession of step 1 1\- sti-p displao-ni'Mits. There may I*- a small ermr in the starting 

 point, !-.</., S(o&) in the alwve example, or in the value adopted for i. We will 

 consider these separately, taking the case where the displacement is on the left, or 

 t he lirst term. 



1. The limit correct, but . s/i</l,tti/ too large. Then S calculated from this is also 

 slightly too large. It will, however, serve to identify a large series of steps in 

 succession, i.e., to reproduce the successive difVen-nr.-s of the wave numbers of the 

 lines. But the errors will all be cumulative, and if the last line of a set be calculated 

 direct from the first, its denominator is too large and its wave number too small. Tn 

 this case a more correct value of S can l)e obtained by using these extreme lines, and 

 this corrected value must satisfy all the other lines. In general a new correction will 

 only affect an extra significant figure in the value of S. 



2. S correct, but limit /n-otig. In this case a slight error in the limit will be of no 

 importance unless the S and its multiples are considerable; and, as a rule, the limits 

 are known with very considerable accuracy, except possibly in the alkaline earths and 

 a few others. Let us suppose the limit adopted (say S()) is too large, that is, its 

 denominator too small. If the second line is due to a positive displacement, its 

 denominator is larger than that of the first, and the wave number less. Suppose D,, 

 D 2 the denominators for the two lines, D 2 > I), if the displacement is positive, the 

 separation is N/D, a N/D/. If the limit is chosen too large D, and D a are chosen too 

 small, although D, D, is correct since S is supposed correct. If D, becomes D, a;, 

 the error in the separation is 8N/Dj' SNor/D,*, which is positive since D a is 

 supposed > D,, i.e., the calculated separation is too large. If the displacement is a 

 negative one, D a < D,, the true separation is now 2N/D/ 2N/D, 2 and the error 

 2NX/D! 3 2NX/D/, which is now negative since D 2 < D,. The effect would be that 

 in any series of step by step displacements S would appear to require continual 

 decreases, and at the end the " corrected values " would not at all fit the initial 

 -rises. If, then, it is found that when S is corrected as in Case 1 the corrections 

 tend to alter the former corrected one, and not to produce additional significant 

 I inures only, it may be surmised that the limit has been wrongly chosen. It is 

 dear, then, that where there are a number of successive collaterals with a large 

 multiple of S between the extreme ones, we have at disposal a means whereby 

 much more accurate values for S and the limits are obtainable. Cases are given 

 below, e.g., in BaD. 



For low atomic weights 5, is always a small quantity and except for orders where 

 i = 1 or 2, the alteration in wave number is small. For the present purpose which 

 is to obtain proof of the existence of the displacements here indicated, no evidence can 

 be admitted in which the change in wave numl>er produced by a displacement <J,, is 

 comparable with the possible error of observation. The evidence, therefore, is of 



VOL. CCXIII. A. 2 X 



