1882.] '^^* [Chase. 



some test which will be rigid enough to fulfill the broadest requirements of 

 mathematical likelihood. If we substitute j\ D, for | D, in the ratio of 

 probability^ to improbability, we provide for requirements of linear oscilla- 

 tion, orbital motion- and gravitating tendency. In such limited ranges of 

 comparison as are possible for the chemical elements, most mathematicians 

 would, perhaps, be satisfied with this substitution. All doubt should be 

 removed by introducing the coefficient of probable error, .674489, and using 

 .674489 X i D = .168622 D. If we let n represent the number of terms 

 in a given group, the ratios of probability, which have been found in Notes 

 171-3, should be multiplied by .674489", in order to give results which are 

 entirely independent of any a priori assumption. We then find 



• For the monatomic group. Note 171 73.75:1 



" trivalent " "' 171 6885.88 : 1 



di-ortetratomic " " 172 21253910000.00:1 



" supp'yartiad " " 172 37337.33:1 



aggregate valency 403(10)'^ : 1 



" metallic group, Note 173 1 : 44.33 



The mean ratios, for single representatives of the several groups, are 

 the following : 



For the monatomic group 1.478 : 1 



trivalent " ' 3.534 : 1 



1st artiad " 4.050 : 1 



" 2d " " 1.550 : 1 



" aggregate valency 2.235 : 1 



" metallic group 1 : 1.145 



The uniform character of the phyllotactic indications, in the groupings 

 which are based upon similitudes of chemical affinity, is very satisfactory. 

 To all who are willing to attach weight to a priori considerations, the 

 following statement of mean ratios may be acceptable : 

 For the monatomic group, Note 171 



" trivalent 



" 1st artiad " 



" 2d " 



" aggregate valency 



" metallic group, 



' ' perissads 



" artiads 



" hydrogen unit 



The last result was quite unexpected. It was obtained by assuming .250 

 as a probable mean difference from exact multiples of H, and treating all 

 the values in Clarke's table in the same way as in the phyllotactic exami- 

 nations of Notes 171-3, so as to obtain, for each element, the ratio, ^ H : 

 (T-0). Although the aggregate evidence of phyllotactic influence upon 

 valency, (3.313 ; 1), is nearly 1.6 times as great as the evidence of 



