150 Prof. H. Hilton on the 



For the longitudinal vibrations we have 



i' 1 =h(-2z 1 + z 2 ), z 2 = Jt(z 1 -2z 2 + z 5 ), . .., 



z n -i = li(z n _ 2 -2zn-i + z n ), z n =h(z n - l — 2z n ), . . (iii.) 



where A=P/M(c— rf). 



Similar equations hold good for the lateral vibrations, 

 if we replace P/M(c-d) by P/Mr. 

 Multiplying equations (iii.) by 



sin nky, sin (n — l)ky, . . . , sin 2ky, sin ky, 

 where (n + l)y=7r, and adding, we have 



"&=— 4fcwn»i*yk no- 

 where 



fj. = sin ?i £7 Z\ + sin (n — 1) ky z 2 + . . . 



+ sin 2ky z n - i + sin ky z n . . . (v. ) 



The quantities f A are the " principal coordinates " (normal 

 modes) of the longitudinal movement. 



The periods of the principal oscillations are 



7tA~2 cosec iky, 

 where 



k = \, 2, . .,n, and (n + l)7 = 7r. 



Equations (v.) give 

 i(n+X)z r = sin (?*-?• + 1)7 fi + sin (n — r + l)2y$ 2 +.. . 



4- sin (n — r + 1) 717 £ n . . ( vi. > 



Hence, i£ the system is performing a principal lateral 

 oscillation in a plane through the string, so that every 

 principal coordinate except one is always zero, the particles 

 lie at any instant on a sine-curve. 



The sine-curve has OZ as axis and intersects it at 0, Z 

 and at 0, 1, 2, . . . , n — 1 other points according as k — l 3 

 2, 3, ... , n. 



As a verification of the above result, the reader may 

 show that, when ?*— ^oo and (n-\- I)c=L-nM//), the periods 

 of the principal lateral vibrations tend to 2Lpi/kTi; and 

 similarly for the longitudinal vibrations. These are the 

 well-known results for the periods of vibration of a tense 

 uniform string (see Lamb's ' Theory of Sound,' § 25). 



§ 3. Suppose, now, that the n particles are at rest in a 

 line with each two adjacent particles connected by a light 

 spring at its natural length c (all the springs being of the 



