is in one plane, but when in space it is ds² = dx² + dy² + dz²: and as dsdt, the element of the space divided by the element of the time is the velocity: therefore,
12 dx² + dy² + dz²dt² = 12 v²;
consequently,
2ƒ(x, y, z) + c = v²,
c being an arbitrary constant quantity introduced by integiation.
77. This equation will give the velocity of the particle in any point of its path, provided its velocity in any other point be known: for if A be its velocity in that point of its trajectory whose co-ordinates are a, b, c, then
A² = c + 2ƒ(a, b, c),
and
v² - A² = 2ƒ(x, y, z) - 2ƒ(a, b, c);
whence v will be found when A is given, and the co-ordinates a, b, c, x, y, z, are known.
It is evident, from the equation beng independent of any particular curve, that if the particle begins to move from any given point with a given velocity, it will arrive at another given point with the same velocity, whatever the curve may be that it has described.
78. When the particle is not acted on by any forces, then X, Y, and Z are zero, and the equation becomes v² = c. The velocity in this case, being occasioned by a primitive impulse, will be constant; and the particle, in moving from one given point to another, will always take the shortest path that can be traced between these points, which is a particular case of a more general law, called the principle of Least Action.
Principle of Least Action.
79. Suppose a particle beginning to move from a given point A, fig. 20, to arrive at another given point B, and that its velocity at the point A is given in magnitude but not in direction. Suppose also that it is urged by accelerating forces X, Y, Z, such, that the finite value of Xdx + Ydy + Zdz can be obtained. We may then determine v the velocity of the particle in terms of x, y, z, without knowing the curve described by the
