HARMONIC ANALYSIS. 



79 



devise a method of separating them out of the mixed results of a harmonic 

 analysis. Hermann* gives the following rules: (1) When a harmonic of 

 large amplitude appears with neighboring harmonics of very small ampli- 

 tude, it may be considered alone as indicating approximately the partial. 

 (2) When the strong harmonic is accompanied by two neighboring strong 

 harmonics, all three should be considered. (3) When one of the neighbor- 



10 

 9 

 8 

 7 

 6 

 5 

 4 

 3 

 2 

 I 



ro 

 9 



8 

 7 

 6 

 6 

 4 

 3 

 2 



12 

 II 



)0 

 9 

 8 

 7 



6 



S 



4 



3 



i 

 I 



10 

 9 

 8 

 7 

 6 

 5 



4 



3 

 Z 



12 3 4 5 



I 2 3 4 S 



JJ 



10 

 9 



8 

 7 

 6 

 5 

 4 

 3 

 2 

 I 



12 3 4 5 



12 3 4 5 



12 3 4 5 



Fig. 77.— Plot of Fio. 78.— Plot of five Fig. 79. — Har- Fio. 80.— Plot of five Fio. 81. — Har- 



five harmonics. harmonics with one monic plot to harmonics with one monic plot to 



inharmonic. figure 78. inharmonic. figure 80. 



ing harmonics is more than twice as great as the other, only the former 

 should be considered with the strong harmonic. In the case, for example, 

 where the first 10 sinusoids in the result have the relative amplitudes 4.2, 

 8.5, 3.2, 7.3, 2.2, 13.9, 44.7, 50.2, 13.6, 14.6, Hermann calculates the relative 

 period of the upper inharmonics from the four strong harmonics. The 

 weighted mean is 



(13.9 X 6) + (44.7 X 7) + (50.2 X 8) + ( 13.6 X 9 ) _ ^ ., 

 13.9 + 44.7 + 50.2+13.6 



That is, the frequency of the inharmonic is 7.53 times that of the funda- 

 mental. In this particular case the latter was 98; consequently the 

 frequency of the inharmonic was 737. 



* Hermann, Phonophotographische Untersuchungen, iv, Arch. f. d. ges. Physiol. (Pfliigcr), 1893, 

 Liii, 50. A summary of Hermann's mathematical methods and of the discussion with Pipping is to be 

 found in Burkhardt, Entwickelungen nach oscillirenden Funktionen, Jahresbericht d. deutschcn Matbe- 

 matiker-Vereinigung, 1901-02-03, x, 279-288. 



