Page:Somerville Mechanism of the heavens.djvu/97

Chap II.] space moved over in an element of time, and the element of velocity to be the velocity that a particle would acquire, if acted on by a constant force during an element of time. Thus, if t, s and v be the time, space, and velocity, the elements of these quantities are dt, ds, and dv; and as each element is supposed to express an arbitrary unit of its kind, these heterogeneous quantities become capable of comparison. As a decrement only differs from an increment by its sign, any expressions regarding increasing quantities will apply to those that decrease by changing the signs of the differentials; and thus the theory of retarded motion is included in that of accelerated motion.

68. In uniformly accelerated motion, the force at any instant is directly proportional to the second element of the space, and inversely as the square of the element of the time.

Demonstration.—Because in uniformly accelerated motion, the velocity is only assumed to be constant for an indefinitely small time, v = $ds⁄dt$, and as the element of the time is constant, the differential of the velocity is dv = $d²s⁄dt$; but since a constant force, acting for an indefinitely small time, produces an indefinitely small velocity, Fdt = dv; hence F = $d²s⁄dt²$.

69. The general equation of the motion of a particle of matter, when acted on by any forces whatever, may be reduced to depend on the law of equilibrium.

Demonstration.—Let m be a particle of matter perfectly free to obey any forces X, Y, Z, urging it in the direction of three rectangular co-ordinates x, y, z. Then regarding velocity as an effect of force, and as its measure, by the laws of motion these forces will produce in the instant dt, the velocities Xdt, Ydt, Zdt, proportional to the intensities of these forces, and in their directions. Hence when m is free, by article 68,

for the forces X, Y, Z, being perpendicular to each other, each one is independent of the action of the other two, and may be regarded as