212 Dr. A. C. Crehore on the Theory of the 



where h is a constant damping factor, and 



T 

 a 2 = — = a constant. (11) 



P ' 



T x being the tension and p the mass of string per unit of 

 length. 



Instead of using this form of the equation it is advanta- 

 geous to express it in terms of the so-called normal * 

 coordinates, which gives the equation the form 



2 



(f> s + k(f> s + n s 2 <j> s =]-&* (12) 



where k is the air damping factor, and 



n g —— j-; orn/=-^ — , s being an integer. . (13) 



The deflexion of the string, y, in terms of the normal co- 

 ordinates is 



, . irx , , . 2ttx , , . Zirx ,., .. 



y = ^ 1 sin-r- + $ 2 sin — ^ l-^gsin— ^— + (14) 



and the velocity 



. 7T t V • . 2tT# • . Zirx .. K . 



y = cj> 1 sin-r- -f<£ 2 sm — , f-$ 3 sm— |- + (15) 



The customary expression for <3> s in terms of the mechanical 

 force is 



<& g = I p Y sin ^^ dta, 



where pYdx represents the force upon an element of the 

 string. In our case this is replaced by Hidx, where i 

 denotes any variable current, and hence 



<S>, = if Ksm S ™dx (16) 



Assuming that the string is in a uniform field throughout 

 its entire length, this integral is 



3> s = — (1-cosstt) ..... (17) 



As 5 takes in succession all integral values from 1 on, 

 <I>, vanishes for all even values, showing that the impressed 

 force due to a current in the string generates no even 

 harmonics. 



* Thomson & Tait's ' Natural Philosophy/ first edition 1867, § 337. 



