Page:Somerville Mechanism of the heavens.djvu/105

Chap II.] its variation is $$ds.\delta ds = dx .\delta dx. + dy.\delta dy + dz.\delta dz$$ but $$ds = vdt$$,

hence $$v. \delta ds= d \frac{dx}{dt}\delta dx +\frac{dy}{dt} \delta dy+\frac{dz}{dt} \delta dz$$,

which is the value of the second term; and if the two be added, their sum is

$$\delta .v ds =d \bigl\{ \frac{dx}{dt} \delta x + \frac{dy}{dt} \delta y + \frac{dz}{dt} \delta z \bigr\}, $$

as may easily be seen by taking the differential of the last member of this equation. Its integral is

$$\delta\int vds = \frac{dx}{dt} \delta x + \frac{dy}{dt} \delta y + \frac{dz}{dt} \delta z$$.

If the given points A and B be moveable in space, the last member of this equation will determine their motion; but if they be fixed points, the last member which is the variation of the co-ordinates of these points is zero: hence also $$\delta\int vds = 0$$, which indicates either a maximum or minimum, but it is evident from the nature of the problem that it can only be a minimum. If the particle be not urged by accelerating forces, the velocity is constant, and the integral is $$vs$$. Then the curve $$s$$ described by the particle between the points A and B is a minimum; and since the velocity is uniform, the particle will describe that curve in a shorter time than it would have done any other curve that could be drawn between these two points.

80. The principle of least action was first discovered by Euler: it has been very elegantly applied to the reflection and refraction of light.

If a ray of light IS, fig. 21, falls on any surface CD, it will be turned back or reflected in the direction $$\text{S} r$$, so that $$\text{ISA} = r\text{SA}$$. But if the medium whose surface is CD be diaphanous, as glass or water, it will be broken or refracted at S, and will enter the denser medium in the direction SR, so that the sine of the angle of incidence ISA will be to the sine of the angle of refraction RSB, in a constant ratio for any one medium. Ptolemy discovered that light, when reflected from any surface, passed from one given point to another by the shortest path, and in the shortest tune possible, its velocity being uniform.