THEORY OF VIBRATIONS 47 



case of the gravest natural mode the frequency thus obtained 

 will be an upper limit. Take, for instance, the case of three 

 equal particles attached at equal intervals to a tense string 

 ( 14), and consider an assumed type of symmetrical vibration 

 in which x = z \y. The kinetic energy is then given by 



2T = M(d? + y*+ &) = M(I + 2\*)f, ......... (6) 



so that the inertia-coefficient is M(l + 2X 2 ). For the potential 

 energy we have 



2 , (7) 

 a a 



as is found by calculation of the work required to stretch the 

 string (as in 22), or otherwise. The coefficient of stability is 

 therefore P/a . (4\ 2 - 4\ 4- 2). For the speed ( p) we then have 



P 4X*-4X + 2 



This is stationary for X = + ^ \/2, and the corresponding speeds 

 are as in 14. In this case it was evident beforehand that the 

 assumed type would include the true natural modes of sym- 

 metrical character. 



It is unnecessary for the purposes of this book to discuss in 

 detail the theory of dissipation in a multiple system. The 

 general effect is the same as in 12 ; the free vibrations 

 gradually die out, but if the dissipative forces be relatively 

 small the periods are not sensibly affected. 



17. Forced Oscillations of a Multiple System. Prin- 

 ciple of Reciprocity. 



The theory of forced oscillations is sufficiently illustrated if 

 in 15 (12) we assume that Qi varies as cospt, whilst (? 2 = 0. 

 The equations will be satisfied if we assume that q l and q t both 

 vary as cospt, provided 



qi + (cv-pdu) ? 2 = ft, 



ft, I 

 0. J 



These determine the (constant) ratios of q 1 and q 2 to Q l ; thus 



