274 Mr. B. H. M. Bosanquet on the History of the 



consonance-beat or Smith's beat, and the beats of Helmholtz, 

 which are taken to include the beats arising from overtones 

 and combination-tones. 



The account of the unison-beat is in most respects that 

 common to all authors. But there is a point in which it differs 

 from the carefully-guarded exposition of Smith, which gives 

 rise to a fallacy running throughout the whole paper, though 

 distinct from my principal ground of objection. The whole 

 exposition is founded on effects supposed to be produced by 

 the blows of a hammer. Now this makes a very complicated 

 effect the basis of operations. In the first place, we have in 

 the primary impulse a sound rather than an impulse of the 

 air. This example then involves the question of the produc- 

 tion of notes by intermittent sounds, which belongs to a higher 

 and more difficult part of the subject. Secondly, the impulse 

 selected, regarded as an impulse, is of a very complicated 

 character. It can be shown without difficulty that a series of 

 discontinuous impulses is represented by the sum of an infinite 

 series of impulses following the pendulum-law, the amplitudes 

 of all the terms being ultimately equal. 



Consider more closely the example given by Pole. Suppose 

 A to make eleven strokes of the hammer to ten of B. Then, 

 if at one point the strokes of A and B coincide, after five 

 strokes of B, A will be half a stroke in advance, and the 

 strokes will succeed each other with intervals of half a stroke 

 for a short time; that is to say, the pitch of the resulting sound 

 goes up an octave. Now if two pure tones near each other in 

 pitch produce the unison-beat, the beat consists of alternations 

 of sound and silence very nearly — at all events not of alterna- 

 tions of fundamental and octave. Notes are easily procured 

 of sufficient purity for this demonstration. Consequently 

 Pole's illustration fails to indicate the real nature of the phe- 

 nomenon. 



Smith is careful to avoid this difficulty; and his demonstra- 

 tions are perfectly applicable to pendulum-vibrations, although 

 he makes no assumption as to the forms of the impulses em- 

 ployed. 



The explanation given by Pole of the effect of imperfect 

 coincidences is quite correct ; but it is attributed to Young. 

 Now this explanation was fully attained by Smith, and is the 

 origin of the term " period of least imperfections," which is 

 one of the most perplexing elements of his phraseology. 



In alluding to the disappearance of unison-beats in inter- 

 vals greater than a third, Pole notices Young's theory that 

 the beats pass over (? are entirely transformed) into the Tar- 

 tlni tone. But in speaking of the doubt cast by Helmholtz 



