Numerics of the Elements. 399 



opens up an entirely new field for chemical calculators. Our 

 only existing resource is an average, in each particular case, 

 of all unobjectionable results. In the present memoir these 

 averages have been further subjected to geometrical average ; 

 each one thus connecting the other by a method far more 

 severe than that of arithmetical means. 



As regards the very interesting and perennial discussion of 

 Prout's law, it is clear that the numerics which I have calcu- 

 lated are completely opposed to a theory of whole numbers, 

 or integral multiples, for the great majority of the elements. 

 Such a law can only hold in the few cases where x=0 or oc ; 

 and As, Sb, and U appear to be the only instances of it. 



If we consider the expression for any numeric, 



y = p.l5-15(-9375)*, 

 we shall observe that, while the periodic factor continues to 

 increase, the geometric portion of the equation undergoes no 

 change. If p grew very large, a time would arrive when 

 — 15 ("9375)* would cease to materially affect the value of p 15. 

 It is therefore easy to conceive the existence of an upper 

 limit to our existing system of numerics ; indeed, in the 

 natural order of events, such a limit might occur without 

 making p very large. 



Chemists can hardly fail to be struck with the fact that 

 U = 240 has for eight-and-thirty years of vigorous scientific 

 research remained the highest value on their list. Another 

 very interesting and tempting suggestion is latent in the 

 series I5(*9375) z itself. The sum of all its terms between 

 and cc is exactly 240 ; in other words, this number repre- 

 sents the, total formative change in any given group. Hence 

 240 can hardly fail to be a critical number in, and may very 

 probably be the upper limit of, our existing system.. 



If we accept this suggestion, the general equation for 

 numerics assumes the following very simple character — 



y=^-"(-fi7> 



n + 1 being the number of groups in the system. Hydrogen, 

 which all systematists agree in excluding from the common 

 system, may from the new point of view be regarded as a 

 relic of earlier changes embodied in the equation 



y=lx i-i(ji-J. 



This system would contain only two groups, of the first of 

 which hydrogen would be the terminal member. 

 Glasgow, Oct. 10, 1884. 



