1881.] 611 [Chase 



If the Fraunliofer lines are, like musical beats, due to the interference of 

 ■waves wliicL are very nearly but not exactly in unison, tlieir proper in- 

 vestigation requires a consideration of more intricate harmonies than those 

 ■whicli are based upon simple binary or ternary division. Tlie liarmonic 

 A line seems to be attracted towards tlie simple octave of H.^, and Hj indi- 

 cates a reaction resulting from such attraction. For, if Hj be divided into 

 13 parts, (3 X 4), and H.^ into 15 parts, (3 x 5), each of the submultiples 

 "will also be a submultiple of A, very nearly, if not exactly. 



(1) 



The line bj is very nearly f, or more nearly |f of Hj ; the lines b4, E, 

 D], are very nearly -^l (or about |), |f (or about |), and | of U^, respec- 

 tively. In other words, Hj, bj, A are nearly enough in the simple har- 

 monic ratio 3, 4, 6, to produce luminous beats, or dark lines, while H^, b^, 

 E, Dj, A, show a like approximation to 13, 15, 16, 18, 34. 



Tlie tendency of the harmonic ratios to become simply geometric, (Notes 

 41-43), is illustrated by approximations to the following equations : B = 

 T/aTO ; Dj =^ v'aF ; D^ = l/C E ; l\ = v/cTi ; b, = Vh^^ ; U, = v^hH^. 

 The first of these equations gives the following accordances : 



Harmonic. Observed. Error. 



a =718.51 a 718.47 +.04 j 



IS = V^f = 686.67 B 686.71 —.04 [. (O) 



y =656.25 C 656.31 +.04 ) 



From (1) and (3) the harmonic series in the following table is constructed 

 by using the exact harmonic ratios |, |, |, f, f, together with the disturb- 

 ing ratios ||, f|, |f, (which are all near enough to | to produce luminous 

 beats), and ||- which is nearly equivalent to |. Four of the numbers 

 which appear as terms of the exacVharmonic ratios, (3, 3, 5, 8), belong to 

 the phyllotactic series ; one, (6 =: 3 x 3), is the product of two adjacent 

 phyllotactic factors ; one, 7, is a prime number. 



PROC. AMER. :?fittiiOs. soc. XIX. 109. 3t. printed dec. 31, 1881. 



