In other words, the _beats_ travel forward with the same speed as the constituent waves. And in this case there is no difference between the velocity of the wave-train and the velocity of the individual wave. The above proof may be generalized as follows:— Let the wave-velocity vary as the _n_th root of the wave-length, or let _v_^_n_ = Cλ; and let λ = 2π/_k_ as before. Then— _v_^_n_ = 2πC/_k_, and _vk_ = 2πC/_v_^{_n_-1} = 2πC_v_^{-(_n_-1)} also _k_ = 2π/λ = 2πC/_v_^_n_ = 2πC_v_^{-_n_} Hence _d_(_vk_)/_d_(_k_) = (_n_-1_v_^{-(_n_-1)-1})/(_nv_^{-_n_-1}) = ((_n_-1)/_n_)_v_ or V = ((_n_-1)/_n_)_v_ That is, the wave-train velocity is equal to (_n_-1)/_n_ times the wave-velocity. In the case of sea waves _n_ = 2, and in the case of air waves _n_ = infinity. If _n_ were 3, then V = (2/3)_v_, or the group-velocity would be two-thirds the wave-velocity. NOTE B (see p. 273). Every electric circuit comprising a coil of wire and a condenser has a definite time-period in which an electric charge given to it will oscillate if a state of electric strain in it is suddenly released. Thus the Leyden jar L and associated coil P shown in Fig. 82, p. 271, constitutes an electric circuit, having a certain _capacity_ measured in units, called a microfarad, and a certain _inductance_, or electric inertia measured in centimetres. The capacity of the circuit is the quality of it in virtue of which an electric strain or displacement can be made by an electromotive force acting on it. The inductance is the inertia quality of the circuit, in virtue of which an electric current created in it tends to persist. In the case of mechanical oscillations such as those made by vibrating a pendulum, the time of one complete oscillation, T, is connected with the _moment of inertia_, I, and the mechanical force brought into play by a small displacement as follows: Suppose we give the pendulum a small angular displacement, denoted by θ. Then this displacement brings into existence a restoring force or torque which brings the pendulum back, when released, to its original position of rest. In the case of a simple pendulum consisting of a small ball attached to a string, the restoring torque created by displacing the pendulum through a small angle, θ, is equal to the product _mgl_θ, where _m_ is the mass of the bob, _g_ is the acceleration of gravity, and _l_ is the length of the string. The ratio of displacement (θ) to the restoring torque _mgl_θ is 1/_mgl_. This may be called the displacement per unit torque, and may otherwise be called the _pliability_ of the system, and denoted generally by P. Let I denote the moment of inertia. This quantity, in the case of a simple pendulum, is the product of the mass of the bob and the square of the length of the string, or I = _ml_^2.