282 Representation of the Periodic System of the Elements. 



where the elements of the lower series in each set appear to be 

 more metallic in melting-point and other properties than the 

 elements of the upper series ; the same holds good also for nega- 

 tive elements ; thus we have, for example, for the valency 6 



0, Cr, Mo, — , W, U, 



S, Se, Te, — , 



although here the relations are of a somewhat complicated 

 character. At all events it is clear that for each determined 

 value a of the valency the properties of the elements alter 

 periodically with increasing atomic weight, so that e=f(a,p) 

 represents for each a a wave-shaped or zigzag curve. The 

 question now arises whether we really have two different 

 relationships between e and p in the form e=F(p) and 

 e=f(a,p) ; the true answer can, of course, only be furnished 

 by the theory of the elements to be developed. But if we 

 recall certain problems of theoretical physics, for example, in 

 heat-conduction, sound, and light, we perceive at once that we 

 have there also at least two relations of a similar kind ; the 

 one relationship appears as the integral of a differential 

 equation, the other as a so-called limiting condition inde- 

 pendent of the first ; the two contain an undetermined para- 

 meter, which often appears as a whole positive number (e.g.. 

 in the theory of the vibration of strings). Hence it appears 

 not improbable that the chemical theory of the future will 

 also lead to two relationships, 



e =A a ,p), and e = <l>(a,p), .... (2) 



where a — the valency — a whole positive number plays the 

 part of an undetermined parameter. It is to be remarked 

 that so far we always find <x<8 ; but in any case it is not 

 impossible that certain special conditions of the problem might 

 exclude certain values of a. If we now eliminate a from both 

 equations we obtain the relationship of the form 



that is our curve of fig. 1. 



Now we are in position to take another step. If the rela- 

 tionships (2) are really independent, then e may be eliminated 

 from them ; this leads to the relationship of the form 



*(«,/>) =0, (3) 



which will give for each a a completely determinate series of 

 values p, the values of p thus obtained represent, then, the 

 atomic weights of the actually existing elements. It may 

 well happen tLat the number of the real and positive roots of 

 equation (3) will be finite for each a, hence also the number 



