Page:Somerville Mechanism of the heavens.djvu/102

varies according to same law of the distance and leaving them otherwise indeterminate, it is possible to deduce certain properties of a moving particle, so general that they would exist whatever the forces might in other respects be. Though the variations differ materially, and must be carefully distinguished from the differentials dx, dy, dz, which are the spaces moved over by the particle parallel to the co-ordinates in the instant dt; yet being arbitrary, we may assume them to be equal to these, or to any other quantities consistent with the nature of the problem under consideration. Therefore let δx, δy, δz, be assumed equal to dx, dy, dz, in the general equation of motion (6), which becomes in consequence

Xdx + Ydy + Zdz = dxd²x + dyd²y + dzd²z/dt².

75. The integral of this equation can only be obtained when the first member is a complete differential, which it will be if all the forces acting on the particle, in whatever directions, be functions of its distance From their origin.

Demonstration.—If F be a force acting on the particle, the distance of the particle from its origin, F x/s is the resolved portion parallel to the axis x; and if F', F", &c., be the other forces acting on the particle, then X = Σ.Fx/s will be the sum of all these forces resolved in a direction parallel to the axis x. In the same

manner, Y = Σ.Fy/s; Z = Σ.Fz/s are the sums of the forces resolved in a direction parallel to the axes y and z, so that

Xdx + Ydy + Zdz = Σ.F xdx + ydy + zdz/s = X. FΣ.F sds/s = Σ.F ds,

which is a complete differential when F, F', &c., are functions of s.

76. In this case, the integral of the first member of the equation is ∫(Xdx + Ydy + Zdz), or ƒ(x, y, z,) a function of x, y, z; and by integration the second is 1/2 dx² + dy² + dz²/dt² which is evidently the half of the square of the velocity; for if any curve MN, fig. 19, be represeented by s, its first differential ds or Am is

$\scriptstyle {\sqrt {{\text{AD}}^{2}+{{\text{D}}{m}}^{2}}}={\sqrt {dx^{2}+dy^{2}}}$;

hence, ds² = dx² + dy² when the curve