δ∫µ ds = 0.
(2)
(b) Let the point A be kept fixed, but let B undergo an infinitely small displacement BB′ (= q) in a direction making an angle θ with the last element of the ray AB. Then, comparing the new ray AB′ with the original one, it follows that
δ∫µ ds = µΒq cos θ,
(3)
where µΒ is the value of µ at the point B.
6. General Considerations on the Propagation of Waves.—“Waves,” i.e. local disturbances of equilibrium travelling onward with a certain speed, can exist in a large variety of systems. In a theory of these phenomena, the state of things at a definite point may in general be defined by a certain directed or vector quantity P,[10] which is zero in the state of equilibrium, and may be called the disturbance (for example, the velocity of the air in the case of sound vibrations, or the displacement of the particles of an elastic body from their positions of equilibrium). The components Px, Py, Pz of the disturbance in the directions of the axes of coordinates are to be considered as functions of the coordinates x, y, z and the time t, determined by a set of partial differential equations, whose form depends on the nature of the problem considered. If the equations are homogeneous and linear, as they always are for sufficiently small disturbances, the following theorems hold.
(a) Values of Px, Py, Pz (expressed in terms of x, y, z, t) which satisfy the equations will do so still after multiplication by a common arbitrary constant.
(b) Two or more solutions of the equations may be combined into a new solution by addition of the values of Px, those of Py, &c., i.e. by compounding the vectors P, such as they are in each of the particular solutions.
In the application to light, the first proposition means that the phenomena of propagation, reflection, refraction, &c., can be produced in the same way with strong as with weak light. The second proposition contains the principle of the “superposition” of different states, on which the explanation of all phenomena of interference is made to depend.