Proposition 128

Table of Contents

PROPOSITION 128. PROBLEM

If the body at M (Fig.94) is acted on by two forces, of which the one has the direction MP normal toe the given line AC, and the other truly has the direction MQ parallel to AC itself or normal to BC, to determine the curve AM which the body describes in any medium with resistance due to action of these forces.

SOLUTION

Calling CP = MQ = x, PM = CQ = y, the element Mm = ds; and with drawing mp and mq there is produced Pp = – dx and Qq = dy and ds = ( dx 2 + dy 2 ) . Let the force, by which the body is drawn along MP be equal to P and the force, by which the body is drawn along MQ, be equal to Q, truly the resistance is equal to R and the speed at M corresponds to the height v. Now the forces P and Q are resolved into normal and tangential forces with the help of the perpendiculars sent from P and Q to the tangent Tt ; hence the normal force arising from P is equal to : and the tangential force is equal to : Truly by resolution from the force Q the tangential force is equal to : and the normal force is equal to

…

Therefore the total normal force is equal to : and the total tangential force emerging is equal to : which force is diminished by the resistance R, and from which total the accelerating force produces the motion.

Hence from these with the radius of osculation at M put equal to r , it follows that (866). [p. 463] Hence on eliminating v from these equations the equation arises expressing the nature of the described curve. Q.E.I.

Corollary 1

If the element ds of the curve is placed constant, then the radius of osculation

Therefore with this value substituted there is

Corollary 2.

From the equations solved together it is found that From which, if the relation between P and Q is given, the equation is immediately obtained, for which the curve is given in terms of v alone.

Corollary 3

If the body is always attracted by some force towards the centre C, then P : Q = y : x. Therefore this equation is then obtained : Which on putting y = px results in this equation

Corollary 4

In a vacuum, in which R vanishes and the body is attracted towards the centre, this becomes :

Besides truly there is : [p. 464]

From which this equation is obtained : or by taking Q in place of P this equation :

Corollary 5

If the centripetal force attracting towards C is equal to

Therefore in a vacuum this equation is obtained for the described curve : With B put in place for − Af n and with dx constant, this equation arises from differentiation :

Truly with dp made constant, this equation is produced : On making x = 1q , this becomes : Of these equations although the integration is not apparent, yet the integral is which was found in the previous chapter.

[This has been solved by Paul Stackel in the O.O. : With MC = ( xx + yy ) = z the equation is found (601, 685)

Let then we have or

On substituting thus in order that : there is made : With which equations solved there is obtained : ]

Corollary 6

Nevertheless although this equation : is of second order differentials, yet it is more convenient than the differential equation of the first order in determining the curves [p. 465], which the projected body describes attracted either in the simple ratio of the distances or inversely as the square of the distances.

For in the simple ratio there is n = 1 and 2 Bq3ddq + dp 2 = 0. On making dp = wdq; on dwdq account of dp being constant we have ddq = − w , hence

Hence there is obtained : or (with the meaning of the symbols changed) since B is a negative quantity.

Corollary 7

If n = –2 or the body is attracted in the reciprocal ratio of the distances squared, it is in the vacuum the described curve with B taken negative, as is required. Hence on integrating it becomes : and on integrating again : Of which each curve is the section of a cone; that one indeed an ellipse, yet all of these are embraced.

Scholium 1

In the preceding chapter, in which we presented the motion of bodies in a vacuum, we also determined curves which a body described with a centripetal force either proportional to the distances or inversely proportional to the square of the distances; and it was convenient to find these curves from these laws in the given corollaries. [p. 466] Indeed the methods are maximally different; for there we arrived at algebraic equations from the comparison of circular arcs, here truly by integration an algebraic equation between the coordinates is spontaneously given. Truly this method, although it is more convenient in the two cases already given, yet in other cases it is troubled with difficulties. For in other hypotheses of centripetal force this method indeed is unable to give a differential equation for the curve described, that yet can always be done with equal ease by the other direct method to give the curve described. Yet this should be ascribed to defective analysis rather than to the method, when we may know the integral of the second order differentials [This is referred to as a differential of the differential equation in the text] of the equation that is from the method used in the preceding chapter, truly we may not be able to elicit this integral from the second order equation.

Scholium 2

The problems of reciprocal natures that can be proposed around these forces, this has now been solved in Cor. 2, in which from a given curve, with the resistance of the medium and the speed at individual points given also, the forces acting along MP and MQ are sought which produce this motion. As if the curve AMB is a circle with centre at C and having radius AC = a and the resistance is equal to vc and speed is constant, truly v = b, then it is x 2 + y 2 = a 2 and [p. 467]

In a similar manner, since there are five things that are arrived at in the consideration : truly the two forces P and Q, thirdly the resistance R, in the fourth place the speed at the individual places or v, and in the fifth place the nature of the curve described or the equation between x and y, always three of these can be taken as given and the two remaining are to be found from these. On this account there are ten problems that can be formed from the number of combinations, in which three are taken from five. But so that we are not detained to any extent in working these out, and from which not much can be deduced that is useful, we treat a single problem, in which the medium offers resistance in the square ratio of the speed and the curve described is sought from a given centripetal force.