502 Mr. R. H. M. Bosanquot on the Beats 



This is the length of Smith's beat, or of the beat as given 

 by a well-known formula. This is easily verified as follows: — 



88. Let v be the velocity of sound corresponding to wave- 

 lengths a and V, and M, N the corresponding frequencies ; 

 then 



\ m > %' *> 



and q\'=p\ + § becomes (n$ = X), 



H" "" M wM ' 



which connects the expression above obtained with the ordi- 

 nary formula for the frequency of the beat. Hence the Smith's 

 beat in this case corresponds in period to the projecting bow 



formed by the -th part of the whole periodic curve of slow 



disturbance of one of the vertices. 



E F 



89. Case II., where — = — n so that the condition for a ver- 

 tex reduces to 



. 27TX . 2-7T , x ^ 



sin — 1- sin r-r US — a) — U. 



A A 



This condition gives the following series of values:— 

 X — X + « a A 



A \' > A + V 



_ -X + X + u _ (a + \')k 



^7 ? = A + A/ ' 



— —n + v^' + a __ ( a + ^0^ 



A + V ; 





A/ > 



until 







j/V=A+V; 



and if 







(fk f =p A, 





p + y 





P 



Where this is not a whole number, the condition will be 



