170 V. OSCILLATIONS AND CYCLIC MOTIONS. 



both for free and forced vibrations, and by an electrical application 

 has solved a very important telephonic problem. 1 ) 



On account of the importance and typical nature of the problem 

 of the continuous string, we shall also solve it by means of Hamilton's 

 Principle. Replacing the length of a segment a by the differential dx, 



writing gdx for the mass m, and for yY-of (partial derivative because 



y depends upon both t and x), and for the sum, the definite integral, 

 we have the kinetic energy 



104) r = 



Similarly in the potential energy the limit of the term 



1S 



so that the potential energy becomes 

 105) W 



As the number of degrees of freedom is now infinite we are 

 not able to use Lagrange's equations, but we can use Hamilton's 

 Principle, which includes them. 



106) 



to 



Integrating the first term partially with respect to t and the second 

 with respect to x, 



i t, 



107) 



The variation dy is as usual to be put equal to zero at the time 

 limits, and, as the ends of the string are fixed, dy equals zero at 



1) Pupin, Wave Propagation over non - uniform Electrical Conductors. 

 Trans. American Mathematical Society, I, p. 259, 1900. 



