58 PROF. W. M. HICKS: A CRITICAL STUDY OF SPECTRAL SERIES. 



unconvincing in the absence of any exact knowledge as to the connection of wave- 

 length with atomic weight, even supposing such connection existed. 



Our knowledge of series spectra is chiefly one might say almost wholly due to 

 the sets of very exact measurements of KAYSER and RUNGE, and of RUNGE and 

 PASCHEN, supplemented by extensions to longer and shorter wave-lengths by 

 BERGMANN, KONEN and HAGENBACH, LEHMANN, RAMAGE, and SAUNDERS. These 

 have been only quite recently added to by PASCHEN* and by the remarkable 

 extension of the Sodium Principal series up to 48 terms by WooD.f A most 

 valuable feature of KAYSER'S work was the publication of possible errors of obser- 

 vation. This has rendered it possible to test with certainty whether any relation 

 suggesting itself is true within limits of observational error or not. In fact, without 

 this, the investigation, of which the present communication forms a first part, could 

 not have been carried out. So far as the author knows, SAUNDERS is the only other 

 observer who has accompanied his observations with estimates of this kind. Others 

 have given probable errors practically estimates of the exactness with which they 

 can repeat readings of that feature of a line which they take to be the centre an 

 estimate of little value for the present purpose. In deducing data from a set of 

 lines it is thus possible to express their errors in terms of the original errors in the 

 observations, and limits to the latter give limiting variations to the former. We 

 therefore . know with certainty what latitude in inferences is permissible, and are 

 often enabled to say that such inference is not justifiable. 



The formula? of RYDBERG or of KAYSER and RUNGE are sufficient in general to 

 identify lines as belonging to a given series, except in the cases of lines in the infra 

 red, where the order is 1 or even 2, but they are not sufficiently accurate for our 

 purpose. It is necessary to obtain formula? for the various series which can reproduce 

 the known series within limits of error or, if this is impossible, with as few outside as 

 possible. The fact that RYDBERG'S formula contains two arbitrary constants, whereas 

 K.R.'s have three, and the well-known relationships brought to light by RYDBERG'S 

 formula naturally suggest it as the basis for a second approximation. In the absence 

 of any definite theory as to the nature of the vibrations giving rise to spectral lines, 

 it is necessary to make some assumption and to test it by its results. The most 

 natural one clearly is to suppose that in RYDBERG'S form A-N/(D) 2 where D = ?>i + ^, 

 D is some function of m which can be expanded in a series of which m + p is the first 

 term. Comparison with observation shows that his formula becomes more exact with 

 increasing m, and that, therefore, the function should be expansible in negative 

 powers of m. In other words, 



m m 

 where a, fi are small. 



* ' Ann. d. Phys.,' 27, p. 537. 

 t 'Astro. Jour.,' xxix., p. 97. 



