of Consonances of the Form h : 1. 423 



does not agree with that required for Konig's first beat-note, 

 which has the same speed as Helmholtz's difference-tone, or 

 (p — q) in the above notation. 



11. The relationship of these resultant displacements to the 

 phenomena in the general case, is most conveniently studied 

 by means of the curves drawn by Donkin's harmonograph. 

 The instrument in my possession has a rather restricted 

 number of change-wheels; and one of my first tasks in the 

 St. John's College laboratory has been to cut additional change- 

 wheels for this instrument, for the purpose of illustrating this 

 subject graphically*. (See Plates IV. to VII.) 



12. The examination of the curves leads us to the following 

 conclusions. 



In every case, whether of beats of unisons, or of beats of 

 imperfect consonances, the examination of the curves shows a 

 portion of a harmonic curve lying through the vertices of the 

 single resultant vibrations, which portion corresponds in du- 

 ration to the beats as given either by Smith's rule or the ordi- 

 nary rule for beats. 



The durations of these harmonic curves are different in dif- 

 ferent cases. Three principal types may be distinguished: — 



Let E, Fbe the amplitudes; p : q the ratio in lowest terms 

 of the exact consonance whose small variation is considered 



(1) If E \p is considerably less than F | q, there are q 

 complete harmonic curves both at top and bottom, and the 

 duration of each is q times that of the Smith's beat. 



(2) If E | p = F | q, there are p + q complete harmonic 

 curves which may be called external, passing both top and 

 bottom, and the duration of each external curve isp + q times 

 that of the Smith's beat; also there are q — p internal curves, 

 which lie nearer the middle; the duration of each internal 

 curve is q— p times that of the Smith's beat. 



(3) IfE | p is considerably greater than F | q, there arep 

 harmonic curves both at top and bottom. They are not com- 

 plete, but appear to form portions of curves of long period. 



13. In all cases the curves which correspond to the beats, as 

 ascertained by Smith's method or the ordinary formula, lie like 

 series of bows, one series at the top and the other at the bottom. 



The complete period of the pendulum-vibration, of which 

 each of these bows forms a part, is always longer than the single 

 bow or Smith's beat, according to the above rules. 



14. Now, according to a well-known principle of mechanics, 

 no pendulum-vibration can give rise to one of another period, 



* These curves are of such interest that I devote some space to their 

 discussion, § 77 &c.;j 



