THE OK Y OF VIBRATIONS 11 



the time t as abscissa and the displacement x as ordinate is 

 of great value. This is called the " curve of positions," or the 

 " space-time curve." In experimental acoustics numerous 

 mechanical and optical devices have been contrived by means 

 of which such curves can be obtained. In the present case 

 of a simple-harmonic vibration, the formula (2) shews that the 

 curve in question is the well-known " curve of sines." 



6. Further Examples. 



The governing feature in the theory of the pendulum is 

 that the force acting on the particle is always towards the 

 position of equilibrium and (to a sufficient approximation) 

 proportional to the displacement therefrom. All cases of 

 this kind are covered by the differential equation 



and the oscillation is therefore of the type (2) of 5, with 

 n z = K/M. The motion is therefore simple-harmonic, with 

 the frequency 



determined solely by the nature of the system, and independent 

 of the amplitude. The structure of this formula should be 

 noticed, on account of its wide analogies. The frequency 

 varies as the square root of the ratio of two quantities, one 

 of which (K) measures the elasticity, or the degree of stability, 

 of the system, whilst the other is a coefficient of inertia. 



Consider, for example, the vertical oscillations of a n 

 mass M hanging from a fixed support by a helical 

 spring. In conformity with Hooke's law of elasticity, 

 we assume that the force exerted by the spring is 

 equal to the increase of length multiplied by a certain 

 constant K, which may be called the "stiffness" of 

 the particular spring. In the position of equilibrium 

 the tension of the spring exactly balances the gravity 

 Mg\ and if M be displaced downwards through a 

 space x, an additional force Kx towards this position 

 is called into play, so that the equation of motion is of 



