16 Proceedings of the Royal Irish Acadeniij. 



These polynomials may then be chosen so as to make the expansions satisfy 

 certain other conditions. 



Arts. 5-8. Application to Transverse Vibrations of Mastic String whose ends 

 are attached to a System of Particles with Massless Elastic Connexions. 



Akt. 5. Equations to he satisfied : the Actual Solution. 



As an example of the application of such expansions I proceed to consider 

 a generalization of Lord Kayleigh's example referred to above. Let there be 

 a system, which is capable of vibrating, consisting of an elastic string and at 

 each end a number of particles. We may add to the symmetry w^ithout 

 increasing the difficulty by supposing that every particle of the system at 

 either end is elastically connected to every other and also to a fixed point. 

 Let the motions of any or all the particles be resisted by viscous forces of the 

 usual types, i.e. proportional to the actual or to the relative velocities ; but let 

 it be supposed that the string itself is not subject to viscosity. Let there be 

 m particles at one end x = a, and n at the other x = h. It is supposed that at 

 each end the position of one particle coincides with that of the end of the 

 string. 



The initial displacements and velocities are given, and the subsequent 

 history of the system is required. 



Supposing h > a, at the end a we have a special case of a system of 

 m equations of the form 



(MuD' + knD + X„ - Td/dx)Y, + (M„D' + k,,D + \n)Y, 



+ (M,,D' + h,D + An) F, + . . . = 0, (7) 

 {M^,D' + h,D + Aj,) Y, + {M.-D" + /c,J) + A«) Y, 



+ {M,,D' + h;,D + X,,)Y, + . . . = 0, (8) 

 {M„D' + h,D + A„) r, + {M,,D' + h,D + A^OF, 



+ {M,,D' + k,,D + A33) Fa + . . . = 0, (9) 

 where D denotes djdt and the F's denote the displacements of the particles. 

 Actually the system of coefficients of the F's is symmetrical about its leading 

 diagonal, and, when r, s are different, each constant of the type Mrs is zero 

 and those of the types krs, Xr, are negative. We will consider the more 

 general system above, however, as it presents no greater difficulty. It will 

 be convenient to denote by Ao and also by Ea(d/dt, djdx) the determinant 

 composed of the coefficients of the F's, (and in it D may be replaced as far as 

 we please by fic), so that the displacement (/> of the string itself is subject to 

 the terminal equations 



Fa {didt, djdx) <p = 0, at x = a, (10) 



Ft {djdt, djdx) ^ = 0, at x = h. (11) 



