1882.] 423 [Chase. 



The logarithm of aggregate relative probability is TT.3223273 — IT- 

 .3104047 = 2.0119226; the log. of mean relative probability is ¥.8135919 

 — "2 .5900450 = .2235469. Hence the a,ggregate relative probability of the 

 nitrogen divisor, P, -f- Pj == 102.783 ; the mean relative probability, p^ -f- 

 ^1 = 1.6732. 



272. Aggregate and Mean Ratio of Residuals to Atomic Divisors. 



In the following table the logai'ithms for each group are computed after 

 the method of the foregoing note. The divisor for the first surd, Sj, is 



1 (3—1/ 5) = .382 ; for the second surd, S,„ Ki/ 5 — 1) — .GIB ; for hydro- 

 gen, H = 1 ; for Gerber and Chase I, see Note 136 ; for Chase II, see Note 

 269 ; for Chase III, see Note 271. 



Group. Sa- Si. H. Gerber. Chase I. Chase II. Chase III. 



Monat. 8"..3393239 8.5143370 T^.2138579 TT.7186091 TT.628303S TT.4G19386 T^.2138579 



3 and 5. T.4310096 "f. 7210766 TIT .7187995 TT.3223273 TT.3223273 ■?.0673580 T3.3104047 



2 and 4. TT.7492826 TJ.6963491 T6.5909435 ^3-.8649303 2 3.6476134 ^.3406987 ^3.3406987 

 Metal. T8-.5073,145 T^.I410255 ^3.5593397 T8-.3369171 ^0.2251388 ^2.0499751 3 2-.5388218 

 Periss. T^.7703335 T5-.2354136 ^^.9326574 ^T.0409367 ^2.9506311 To".4692966 23-.5242626 

 Artiad. 2T.2565971 3 3.8373746 JF. 1502832 ¥o.2018474 ?3.8727522 4^.3906738 oo. 8795205 

 Aggreg. ^2.0269306 4T.0727882 -^9.0829406 ^.2427841 6I-.8233833 ^"^.8599704 T9.4037831 

 Mean. T.3441708 T.2667623 T.0794209 T.0506685 T.012S654 ^.9978120 ^.7719341 

 Rel.Ag. .0000000 4.9541424 16.9439900 18.7841465 21.2035473 22.1669602 36.6231475 

 Rel. M. .0000000 .0774085 .2647499 .2935023 .3313054 .3463588 .5722367 



The aggregate residual ratio for S,^ is more than 87,900,000,000,000,000 

 times as great as for hydrogen, and more than 4,199,000,000,000,000,000,- 

 000,000,000,000,000,000 times as great as for the relations to H, N, O ; the 

 mean ratio is 1.8397 times as great as for 11, and 3.7345 times as great as 

 for my second group of divisors. The aggregate hydrogen ratio is more 

 ,than 47,770,000,000,000,000,000 times as great as for my third group, the 

 mean ratio being 2.0299 times as great. 



273. Comparison of Oeometric and Arithmetic Residual Means. 



The logarithms of the geometric mean residual ratios, for the several 

 groups, may be found by dividing the monatomic logarithms by 11, the 

 tri- and pentatomic by 9, the di- and tetratomic by 17, the metallic by 27. 

 Some questions of relative probability may he tested more readily by 

 arithmetical means, and for this reason as well as In order to preserve 

 additional evidence of phj^llotactic influence, the following table is given. 

 All of my divisors were deduced from phyllotactic considerations ; the first 

 set shows the great superiority of my phyllotactic over Gerber' s approxi- 

 mately phyllotactic divisors ; the second set introduces corresponding terms 

 of two phyllotactic series ; the third set has two divisors which are simply 

 phyllotactic (1, 8) and two which are products of phyllotactic ratios 

 (TV = iXi;i = ^X I). 



