PROPOSITION 92

PROPOSITION 92. Problem

845. To determine the effect of the pressing force on the body moving on any surface, that is not acted on by any additional forces.

Solution

Because this pressing force is normal to the surface and thus the direction of this is in the direction MN, this affects neither the speed nor the direction, as the whole force is taken up on pressing the surface, and therefore the body progresses on the same line on which it was moving if this force were absent; but this is the shortest line determined in the preceding proposition. Therefore the body is moving on the line Mmμ , and the radius of osculation of this MO lies along the normal to the surface MN. Therefore let the direction of this pressing force be MN, which therefore presses the surface inwards along MN.

This pressing force is put equal to M; by that the surface is pressed on by a force along MN equal to M. But if the radius of osculation MO is put to lie along the same [undirected] line, then the centrifugal force is contrary to the pressing force, and the effect of this is lessened. Since moreover Mmμ is the shortest line, the radius of osculation is (73) : if twice the height v corresponding to the speed at M is divided by which, then the centrifugal force is produced. On this account the force by which the surface is pressed along MN, is equal to

[Note that the radius of curvature has been given as negative above for a concave curve and the sign of this has been reversed, to give a greater force pressing into the curve.] Finally the position of this pressing force has been found previously (68) : AH = x + Pz and HN = −Qz − y , obviously on being sent from the point N, in which the normal MN intersects the plane APQ, with the perpendicular NH to the axis .Q.E.I.

Corollary 1

846. Since no another force is deflecting the normal force, neither a tangential force nor a resistive force if that is present affect the force pressing on the surface, [p. 471], and it is evident from any forces besides acting on the body that the pressing force is always to be of such a size as we have assigned here.

Corollary 2

847. Therefore however great the departure between the path described by the body from the shortest line, the pressing force on the surface is still along the normal to the surface or along the radius of osculation of the shortest line, not along the radius of osculation of the curve described by the body, and neither is the length of this required for the pressing force.

Scholium

848. For that reason we have used that formula of the radius of curvature of the shortest line, in which differentials of the second order are not present, lest these depend on the positions of the two elements Mm and mμ , through which the body is itself moving. But now the radius of osculation must be known from a single element Mm ; for if the body does not describe the shortest line on account of a deflecting force, then differentials of the second order ddy and ddz must be advance, not present in the radius of osculation of the shortest line.

PROPOSITION 93 Problem

849. To determine the effect on the motion of the body of the tangential force that pulls the body along the tangent line MT (Fig.92) on some surface.

Solution

Let this tangential force be equal to T and the body is progressing through the element Mm with a speed corresponding to the height v ; since this force diminishes the motion, then with the quantities maintaining the same denominations that we have used previously. Now besides this force does not affect either the pressing force nor by the deviation from the shortest line. Now according to the position of the direction of this force, the tangent MT is produced that then crosses the plane APQ at T, then T is a point on the element qQ produced.

Therefore and hence From T the perpendicular TF is sent to the axis; then

whereby there is obtained :

Again since dx : dy = zdx

y − FT then FT = y − zdx , from which the point T is dz dz determined. Q.E.I.

Corollary

850. Since resistance is to be referred to the tangential force, from these it is understood, how the effect is to be determined. For if the resistance is equal to R, then