434 Mr. J. H. Jeans on the 



in this way obtaining (2»+l) independent normal vibrations, 

 the frequency o£ each being the same. 



From the manner in which the analysis has been conducted, 

 it will be clear that all possible normal vibrations must have 

 been taken into account. 



The Spectrum of the Ideal Atom. 



§ 16. For the vibrations of the first class /> = 0; so that 

 these vibrations are not. represented in the visible spectrum. 



The frequencies of the second class of vibration are all con- 

 tained in equation (18). Solving for p 2 , this equation can be 

 split up into the separate equations 



^=/ 2 («), &c. 



Any one of these equations gives rise to a simply infinite 

 series of lines in the spectrum, the wave-lengths of the lines 

 in any such series being found by assigning the various posi- 

 tive integral values to n, in a formula in which n alone varies. 

 Such a collection of lines will be referred to as a " spectrum- 

 series." It has accordingly been shown that the visible 

 spectrum of the ideal atom can be sorted out into a collection 

 of spectrum-series ; and it has also been shown that the ?ith 

 line in each series must be regarded as the superposition of 

 of (2n+ 1) equal free periods. 



Heads of Sjject rum-Series. 



§ 17. It can be shown that every spectrum-series tends to 

 a definite limit corresponding to ?* = co, and the position of 

 the line n = =c will accordingly be referred to as the " head M 

 of the series. 



The heads of the various series will be given by (cf. 

 equation 18) 



/( i A»)=0, (19) 



this equation being the result of eliminating the four variables 

 from equations (13) to (16) after putting ?* = co . 



Let us begin with the consideration of the form assumed 

 by equation (15) when n = co . We have 



00 



%1 (R)=XT n P n (cosy), 







where 



lis r n 



P = /go; , n = ^^+i P »( C0S ?)> 



K vV* — 2rr cos y + r 2 o? 



