Proposition 78

Table of Contents

PROPOSITION 78. Problem

In a medium with some kind of resistance and with some kind of forces acting to find the brachistochrone AM (Fig.78), upon which a descending body arrives the fastest from A to M.

Solution

Let A be the starting point of the motion, through which some straight line AP is drawn being taken for the axis, on which the abscissa AP = x it taken; to which there corresponds the applied line PM = y and the arc AM = s. Again let the speed of the body at M correspond to the height v and the resistance depending somehow on the speed equal to R.

The body is acted on by some absolute forces act on the body, in place of these two forces can be substituted in the given directions ML and MN, of which the former is parallel to the axis AP, and indeed the latter normal to that axis.

The force acting on the body following ML is equal to P and the force along MN = Q. From these forces, there arises the equation : dv = Pdx + Qdy − Rds.

The nature of the brachistochrone curve gives …

(673) with r denoting the radius of osculation of the cure at M towards the upper

ds , since the other direction; hence from which on taking dx as constant we put dxddy 3 − ds . Hence from these two equations : direction must give

…

if v is eliminated, the equation is had for the brachistochrone curve sought ; clearly there is given

…

with the differential of this substituted in place of dv and in place of v itself in the resistance R, the equation is given for the curve sought. Q.E.I.

Corollary 1

The equation for the curve, if v is eliminated in the stated manner, becomes a differential of the third order. Whereby if the threefold integration is to be obtained, then also three constants can be added, in which to be effective, so that on x vanishing, likewise both y and s or v vanish and in addition the curve passes through the given point M .

Corollary 2

Therefore since the brachistochrone curve can always be shown, which has the starting point A and passes through a given point, an infinite number of brachistochrone curves can be drawn from the point A.

Corollary 3

With the equation for the brachistochrone curve AM known, likewise the speed of the descent on that curve can be found at individual points; and as much as …

Corollary 4

With the speed given, from that the time can be determined in which the body completes the arc AM ; clearly the time to pass along the arc AM is equal to : and that can now be found on account of the equation between x and y, even if it has to be shown by quadrature.

Corollary 5. [p. 378]

Therefore if the curve is to be found, which all the brachistochrones drawn from A must cross at right angles, then the construction of this line is obtained, if an amount of the same magnitude is cut off from all the brachistochrones. According to this, the arc of the isochrone is cut off from this infinitude of curves which, since all the curves are brachistochrones, terminate at right–angles to the trajectory. [Thus, the beginnings of sets of orthogonal curves is set out.]

Example

Let the resistance be proportional to the square of the resistance and let the exponent of the resistance be some variable q; then R = qv . Hence since …

on integration it becomes :

But since

…

then the above equation becomes : in which equation v is no longer present. Yet meanwhile this equation becomes an equation of the third order, if the integration signs are removed by differentiation; besides the indeterminate values of P, Q, and q are reasons by which the equation is less able to be prepared for construction.

Scholium

Here it is clearly evident, concerning the brachistochrone curves that have been deduced from the two forces P and Q, since however many forces are acting on the body, all of these can be resolved in this manner into two, but only if the directions of all the forces lie in the same plane.

On this account also in this proposition, the brachistochrones are held for any hypothesis of the centripetal forces acting, which moreover, since neither well–ordered nor constructible equations are forthcoming, we shall not pursued further. Therefore with these disposed of, in which the speed is prescribed by a certain rule, we progress to the following questions, in which curves are required, which sustain a given force on these from the body in motion.