164 V. OSCILLATIONS AND CYCLIC MOTIONS. 



9) 



that is in a normal vibration all the coordinates of whatever sort 

 are in ratios constant throughout the motion, or the solutions 74) 

 are reduced to a single column. The motion is then completely 

 periodic, all the coordinates passing through the equilibrium values 

 simultaneously. We may thus describe the general motion as the 

 resultant of n normal oscillations. Thesjoormal coordinates have the 

 property that the energy of any vibration is the sum of the energies 

 of the separate normal vibrations, for substituting 78) in 76) we have 



80) T+W=( Ci A* + c,AJ + - .+ e.AJ). 



46. Vibration of a String of Beads. Continuous String. 



As an example of the preceding theory let us consider the problem, 

 solved by Lagrange 1 ), of the motion of a string on which are 

 fastened a number of beads of equal mass equidistant from each 

 other and from the ends of the string, the mass of the string being 

 neglected in comparison. Let the number of beads be n, the mass 

 of each, m y the distances apart, a, and the length of the string, 

 I = (n + l)a. Suppose for simplicity the motion of each bead takes 

 place in a straight line at right angles to the stretched string, all 

 the displacements y r being in the same plane. Then the kinetic 

 energy is 



81) T_!5(j, i i + j, ; + . +yi'). 



The coefficients of inertia are the same for all, equal to the mass of 

 any bead. The displacements being small quantities, the length of 

 the string connecting any two beads is equal to a plus small 

 quantities of the second order which will be neglected. The tension 

 of the string will thus be considered constant and equal to S. 

 Neglecting the weight of the beads the only forces acting on a bead 



are the compon- 

 ents of the tension 

 of the two ad- 

 jacent portions of 

 Fig< 36 the string in the 



direction of the 



displacement y, Fig. 36. To find the component we have to multiply 

 the tension by the cosine of the angle made by the displacement 



1) Lagrange, Mecanique Analytique, Tom. I, p. 390. 



