Cbase.] 590 [Nov. 4, 



Tlie ratio between a aud v is between f f and f 3. 



a-^r 



= a. = 1.511936 



62 -f- 41 = &5 = 1.512195 

 65---43 = C5 =1.511628 



&5 — Cft =(?5= .000567 



J- — aj:=:e.. = .000259 



^5 -^ f^o = f 1' tlie probable error being only =h ff- The test, therefore, fails 

 again, the number of failures in the whole comparison being equal to the 

 number of confirmations. Hence it is evident that Schuster's criterion is 

 insufficient, at least when the probable errors of observation are not satis- 

 factorily ascertained. Even if the probable errors were known, the proper 

 application of the test would require supplementary calculations of such 

 intricac}^ as to make it practically inoperative. 



146. Modifications of ilie Test. 



Bj^ increasing the magnitude of the harmonic ratios the test may some- 

 times be made to indicate a probability. For example, /? -v- j' is between 

 I and |. These values give d^ = .033333 ; e^ = .000184 ; e^ h- (Z^ = sMI s- 

 which is less than ^ of the probable error. In like manner c- ~- y'ls, be- 

 tween f and |. These values give d^ = .166667 ; e-^ — .011936 ; e-^-- dr, 

 = tW/s' which is less than -^-^ of the probable error. These results seem 

 to indicate the propriety of harmonic comparisons between terms which 

 are unquestionably of the same order of magnitude. Thus in Schuster's 

 calculation (loc. cit., p. 338), the ratio .96476 lies between |f and ff, the 

 diiference between these two fractions being .016636. The difierence of the 

 fraction in the sodium spectrum from the nearest of these comparative 

 fractions is .000152, which is only .00914 of the difference between the 

 fractions themselves, or less than -^j of the probable error. 



If a supposed harmonic relation can be represented by a fraction with 

 terms of a single digit, Schuster's test might fail even with the above modi- 

 fication, provided the probable error should be > | x i X i ; if the terms 

 are of two digits, it would not be trustworthy if the probable error was 

 > i X 9^ X 9V- If the» modifications of ws viva in synchronous wave sys- 

 tems are of the same order of magnitude as the variations of planetary ec- 

 centricity, the limit of probable error would be at least |, instead of j, of 

 the difference between adjacent fractions which have a common numerator. 

 This would be the case for each of the compared pairs of wave lengths, 

 the probability for the entire system being equivalent to the product of 

 all the independent probabilities. 



All estimates of abstract probability, in such cases, should be greatly in- 

 creased by the a priori probability, or even the mathematical necessity, 

 that synchronous undulations in elastic media must be harmonic. In view 

 ot this consideration, the indications of a harmonic tendency pervading 

 an entire system, such as I have pointed out in many of my compari- 



