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 parallal 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 + dzd²zdt².
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 xs 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.Fxs will be the sum of all these forces resolved in a direction parallel to the axis x. In the same manner, Y = Σ.Fys; Z = Σ.Fzs 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 + zdzs = X. FΣ.F sdss = 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 12 dx² + dy² + dz²dt² 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
;
hence, ds² = dx² + dy² when the curve