282 On the Theory of the Beats of Mistuned Consonances. 



in 1876. If the threads are not so arranged that the pull on 

 the excited vibrator is strictly proportional to the displacement 

 of the exciter, it is quite certain, both from theory and expe- 

 riment, that transformation must take place. My apparatus 

 consisted of a number of pendulums arranged as harmonics to 

 a large and heavy metronome. The connexions were made 

 with elastic threads. When the connexions were so made as 

 to satisfy the above requirement as nearly as possible, I could 

 reduce the excitation of the harmonic pendulums to a very 

 small amount: I never succeeded in entirely stopping it. But 

 it was obviously impossible to fulfil the above condition with 

 any approach to real perfection. When, however, the approxi- 

 mate fulfilment of the condition was purposely avoided, as by 

 letting the thread just go slack at one point of the vibration, 

 transformation set in at once, as it should do, and the small 

 pendulums were set in violent vibration. At that time I 

 pointed out the defect of Mayer's arrangement of transmission 

 by threads (note, p. 865), as it did not appear from the account 

 that any means were taken to render the pull strictly propor- 

 tional to the displacement of the exciter, and consequently 

 transformation was to be expected. 



On the whole, no doubt, the truth of the matter is best 

 stated in a form that combines much of what both parties to 

 this discussion have maintained. It is quite true that, in a 

 hypothetical system in which the forces called into action are 

 strictly proportional to the displacements, the fundamental 

 harmonic vibration cannot permanently excite its multiples. 

 This is unassailable as matter of mathematics; and as to expe- 

 riment, we can only say that, the nearer we approach to the 

 construction of such a system, the less are the multiples excited 

 by the fundamental. But I think that the actual construction 

 of such a system is impossible. And so far as our actual sys- 

 tems depart from the above condition, more or less, transfor- 

 mation does and must take place. Wheatstone's law is there- 

 fore generally true of actual systems ; and it is only incom- 

 plete because it omits the question of quantity. How much 

 of the multiple vibration is excited in systems whose forces 

 depart to known extents from the simple law of proportionality ? 

 So far as we are able to answer by our general knowledge of 

 the facts, they are entirely in accordance with the theory. The 

 more minute comparison of the different classes of systems 

 should be an experimental study of great interest but no theo- 

 retical difficulty, which would materially assist to throw light 

 on the general comprehension of the subject, 



