431 



[Chase. 



288. Another Basis for Estimates of ProbahiUty . 



The substitution of § C for ^ O or 8 H, in Notes 271-3, not only introduces 

 another evidence of phyllotactic influence upon atomicit}^ but it also shows 

 that the organic elements, C, H, O, N, stand in important phyllotactic 

 relations to four fundamental groups of elements. ' If we omit C from the 

 comparison, the remaining elements of the di- and tetratomic group give 

 2". 47682 for their logarithm of mean residual. The respective residuals 

 themselves are .03538Z) and .02998Z>. I have already considered various 

 probabilities, which were based on Schuster's estim'ates, as well as rela- 

 tive probabilities which are independent of any absolute estimates. An-' 

 other satisfactory basis of comparison may be found in the mean limiting 



value of the residual, a'-.nl) = ~ ao,-a (.2-^1)) ^, when the possible re- 

 siduals are taken in arithmetical progression. If all possible values are 

 thus taken, in other words, if the number of terms is infinite, the second 



factor becomes unity and the limiting value is - Aon-A = .183942). This 



is 5.2 times as great as the first of the above mean residuals , or 6.135 times 

 as great as the second. 



289. Resume, of Phyllotactic Atomicity. 



The most satisfactory phyllotactic divisors for the four elementary 

 groups, as indicated by the foregoing notes, are the following : «, for the 

 monatomic group, £"= 1 == |.| ; ,3, for the tri- and pentavalent group, 

 ^.i N = 1.558 ; r, for the di- and tetratomic group, | C'=^0 = 8S = 

 7.9824; d, for the residuary metallic group, 4.^ = | z' = .998. The 

 comparative residual percentages, as deduced from Note 272, and from 

 these divisors are given in the table below : 



290. Comparative Probabilities. 



The following tables give the comparative probabilities for the several 

 divisors : 1st, if the hydrogen unit ; 2d, if .18394i> is taken as the unit of 

 probability. 



