THEORY OF VIBRATIONS 13 



if it be restricted to displacements in the vertical plane through 

 A, B, for the configuration may be specified by the inclination 

 of any one of the strings to the horizontal. Again, the con- 

 figuration of a steam-engine and of the whole train of machinery 

 which it actuates is defined by 

 the angular coordinate of the 

 flywheel. The variety of such 

 systems is endless, but if we 

 exclude frictional or other dis- 

 sipative forces the whole motion 

 of the system when started ^7 



anyhow and left to itself is 

 governed by the equation of energy. And in the case of 

 small oscillations about stable equilibrium, the differential 

 equation of motion, as we shall see, reduces always to the 

 type 6 (1). 



We denote by q the variable coordinate which specifies 

 the configuration. As in the case of Fig. 6, this may be 

 chosen in various ways, but the particular choice made is 

 immaterial. From the definition of the system it is plain 

 that each particle is restricted to a certain path. If in 

 consequence of an infinitesimal variation Bq of the coordinate 

 a particle ra describes an element Ss of its path, we have 

 8s = a$q, where a is a coefficient which is in general different 

 for different particles, and also depends on the particular 

 configuration q from which the variation is made. Hence, 

 dividing by the time-element St, the velocity of this particle 

 is v = adq/dt, or in the fluxional notation *, v = aq. 



Hence the total kinetic energy, usually denoted by T, is 



T=&(m*) = la#, (1) 



where a = 2 (ma 2 ), (2) 



the summation X embracing all the particles of the system. 



The coefficient a is in general a function of q ; it may be 

 called the " coefficient of inertia " for the particular configura- 

 tion q. For example, in the case of the rolling cylinder referred 



* The use of dots to denote differentiations with respect to t was revived by 

 Lagrange in the Mecanique Analytique (1788), and again in later times by 

 Thomson and Tait. We write q for dqjdt and q for 



