ON CONSTRAINED MOTION IN GENERAL.

PROPOSITION I.

Theorem.

12. A body or a point, which moves upon a given line and is acted upon by no forces, will perpetually preserve the same speed: provided that no two contiguous elements An "element" refers to an infinitesimal or infinitely small segment of the path. of that line anywhere form an angle of finite magnitude.

Demonstration.

Because the body, while it moves on the line A M, is acted upon by no force, nor is any place allowed for friction, the motion of the body cannot be changed in any other way except insofar as the line A M prevents the body from being able to move freely; from which, we must investigate what change of speed ought to arise. Let the speed which the body has at M be $= c$; with this speed, therefore, the body would proceed along the tangent M $ν$ original: "tangente M $ν$"; Euler uses the Greek letter nu ($ν$) to mark a point on the tangent line. if it were moving freely. This, however, cannot happen because the body cannot leave the curve A M: instead, the body is compelled to proceed through the segment M $m$. For this reason, let the motion of the body along M $ν$ be conceived as being resolved into a motion along M $m$ and a motion along M $n$, where M $m ν n$ is a rectangular parallelogram A rectangle used for vector decomposition.. It is clear here that the motion along M $n$, the direction of which is normal Perpendicular to the path. to the element of the curve M $m$, is entirely absorbed, and can have no effect on changing the speed. The body, therefore, will proceed with the other motion along M $m$, with a speed which is to the original speed as M $m$ is to M $ν$: therefore the speed with which the body describes the element M $m$ will be $= \frac{Mm \cdot c}{M ν}$. Because, however, M $ν m$ is a triangle with a right angle at $m$, and therefore M $m <$ M $ν$, the speed...