On the Vihrations of Musical Strings. 333 



Cut off and weighed ', and thereby the weight of a yard 

 more accurately ascertained. 



Dr. Smith in the 2d edit, of his Harmonics, prop. 24. 

 coroll. 1. gives this equation: 



35.5 /P 39-126 , . p • ., • 



iYja/ — X - ^ ■- = the number or semi-vibrations made 



in a second ; consequently ■ — -^-^4 / — x ^' = 



' ^ ■ 2xJl:iV / AB 



the number of complete vibrations; wherein P= the 

 weight by which the vibrating string is kept in tension, 

 p = the weight, and AB the length (in inches) of the vi- 

 brating part of it. 



Let the number of complete vibrations be called n ; 

 then, if P be made equal to a given length of the string, 

 as m yards ; and if ?/ = the weight of a yard long of it, 



P will become = irnj, and p = —~-; for AB : /> : : 36, 



(the inches in a yard) : y. Substituting these values of P 

 and p in the equation above in their places, it will become 



355 / 36 wv 39-126 3-1416 / 



''= 2^aT3-V AB^'^-aF=^AB-\/3^'"^39*^26, 



, .„ 3-1416 r-: 



and ABxw= — - — a/ 36»2 x 39-126. 



Now, let m be made = 7000, that is P = as many 

 pounds weight (avoirdnpoise) as a yard long of the string 

 weighs grains troy (^000 gr. being equal to a pound): then 



■11 K-n 3-1416 /~ ■ 



will AB X 7Z = — 5—4/ 36 X 7000X39-126 = 4932'34. 



In words : If tbe string be kept in tension by a weight 

 rqual to 7000 yards long of the string, then will ihe 

 length of the vibrating part (in inches) niulti()lied into the 

 number of vibrations per second, be equal to the constant 

 number 4932-34. 



Although Dr. Smith recommends (p. 15, art. 9.) a cer- 

 tain mode of tuning instruments liavmg the imperfect, 

 scale, which was then and still continues in common use, 

 by making the major thirds and filths beat equally quick 

 with their base, which may suffice for organs, in which 

 the beats are so distinct, lie has not given ^a table showing 

 the relative lengths ot the monochyrd, which would give 

 . the«ounds according to that mode, as he has done at page 



224. 



