Proposition 75

PROPOSITION 75. Problem

663. Between all the curves that join the points A and C (Fig.75), to determine that curve AMC, upon which the body descending from A to C acquires the maximum speed with the resistance present as some power of the ratio of the speeds, and with the uniform force acting downwards.

Solution

In order that the body can arrive at the point C with the maximum speed, each two elements of the curve sought AMC Mm ,mμ thus have to be put in place, so that the body by running through these can take the maximum increment of the speed.

For if the body should acquire a greater increase in the speed by passing through the elements Mn ,nμ in other ways, it will also have a greater speed at C.

Therefore by the method of the maxima the position of the elements Mn ,nμ can be found, if these elements Mn ,nμ are compared with their neighbouring elements and the increases of the speed, which is generated in each are put equal to each other. The applied lines MP ,nmp and μπ are drawn in accordance to the vertical axis, and let the elements Pp , and pπ be equal. Also the vertical lines MF and mG are drawn and on the curves the normal elements mf and ng. Now let the force acting be equal to g, the exponent of the resistance be equal to k, with the resistance in the ratio of the 2mth power of the speeds, and the height corresponding to the speed at M be set equal to v. With these in place, the increment of v along Mm is equal to …

and the increment of the height corresponding to the speed, while the body advances along mμ , is equal to …

Therefore while the body performs the elements Mm and mμ , the height v takes an increase equal to …

But the elements Mn and nμ on being traversed take an increment in v equal to : From which equations put in place there is obtained : [p. 358]

Now we have Mn − Mm = nf and

and

Now on substituting these values, this equation arises : But [since Mn = Mm + mg, and mμ = mg + gμ , ] and on account of the similar triangles nfm, mFM and mgn, μGm With these substituted and divided by mn there is produced : The first two members of this equation are differential of the first order, now the third is equivalent to a differential of the second order, that can be rejected, and hence the equation becomes …

or From which equation the position of the elements Mm and mμ are determined. [p. 359]

Moreover in order that we may employ symbols, let AP = x, PM = y and AM = s; then Pp = pπ = dx , mF = dy and Mm = ds and the equation is produced : Now the canonical equation is : m dv = gdx − v mds , k in which if in place of gdx there is substituted from the above equation : then there is obtained : or Let dy = pds and v1− m = u ; and it follows that, from which by integration there is produced : 1 From this u can be obtained, and in turn v = u 1−m , which value substituted in the above equation gives the equation between p and s and consequently between y and s. Moreover it is expedient that the computation be established in this manner towards the construction of the curve. On placing dy = pds these two equations are obtained

..

and

..

From the first equation it follows that which value substituted in the second equation gives : This equation divided by v m+1 p 2 is made integrable and the integral is : [p. 360] or On account of which and and From which equations the curve sought can be constructed easily. Q.E.I

Corollary 1

664. If the radius of osculation of the curve at M directed towards the axis is called r, then [see diagram here : note that this diagram is slightly inaccurate, as the angle increment should be negative, as the body moves down the slope and not up as assumed. Note also that Euler’s geometric derivatives are the geometrical ratios corresponding to the expansion (1 + d ) 2 y = 1 + 2dy / dx + d 2 y / dx 2 ] : dy d . ds = − dx . r

With this value substituted into .. there is obtained: …

2rv is the centrifugal force of the body on the curve in this motion, the direction of gdy which is away from the axis, and ds is the normal force. Whereby in the curve sought the centrifugal force is acting in the opposite direction to the normal force and is in the ratio to the normal force as 2m to 1, that is as the exponent of the power of the speed to which the resistance is proportional to unity.

Corollary 2

665. Therefore all these curves with a concave part are directed downwards. For since the direction of the normal force and of the radius of osculation are considered to be tending towards the same place, the concavity of the curve must also be considered to be downwards.

Corollary 3

666. In a medium with resistance in the simple ratio of the speeds we have 2m = 1. Therefore in this case the centrifugal force is equal and opposite to the normal force. On account of which the curve sought satisfying the trajectory is that described by the body projected freely.

Corollary 4

667. Because in the equation the indeterminates have been separated, three particular solutions can be obtained thus. The first gives the equation αp + (1 − pp ) = 0 , in which case the speed becomes infinite and the equation is satisfied by some line. In the second case p = 1, or dy = ds, which is for a horizontal straight line, and in the third case p = 0, for a vertical straight line; which has this property, that the body descending along it always accrues the maximum increase in the speed.

Example 1

668. The medium resists in the simple ratio of the speeds ; then it follows that m = 12 .

That is taken from the three equations found, which contains dy; it becomes the integral of which is But as then the equation becomes : or on neglecting the constant C, which does not change the curve, then we have : Which equation divided by y and integrated anew gives …

Which is the equation for that logarithmic curve itself, as we found the trajectory under this hypothesis of the resistance in the first book (889).

Example 2

668. Now let the resistance be in proportion to the square of the speeds; then it is the case that m = 1. This equation is taken : Moreover, the integral of this is Hence it becomes : Which is the equation for the curve sought, which has this property, that the centrifugal force of the body is twice as great as the normal force. Therefore the curve is constantly pressed upwards by a force equal either to the normal force, or to half the centrifugal force.

[There is the continual problem with Euler’s mechanics that he does not restrict himself to the forces acting on the body alone, but also includes these reaction forces acting on the curve. Thus, what Euler calls the centrifugal force exerted on the curve is the reaction of the centripetal force, which in turn is the contact force due to the curve acting on the body; again, the normal force is the reaction force of the curve corresponding to the normal component of the weight.]

The body thus moves on this curve so that the height corresponding to the speed at M is equal to :

Scholium 1

670. Although according to any hypothesis of the resistance, there is a place for the particular ratio between the centrifugal force and the normal force, the vacuum moreover should be considered as well as the case for each resistance, and it follows in vacuo that any curve is satisfactory [i.e. as conservative forces only apply.] Also all the curves in vacuo have this property, that upon these curves from the equality of the heights equal speeds are generated, and thus nothing extra can be defined that provide satisfaction to the question rather than the rest of the equations.

Scholium 2

671. In all these curves found that the speed of the body is nowhere equal to zero.

Therefore the problem cannot be resolved by this method, that among all the descents made from A to C from rest, this curve in which the body reaches the maximum speed can be determined ; to which question only the vertical straight line passing through C and joined to the horizontal drawn through A joined together is satisfactory. Moreover our solution has been prepared, so that the positions of any two neighbouring elements can thus be defined, which produce either the maximum or minimum increase in the speed. On account of which by this method that curve is found, upon which the motion of the body acquires either a greater or smaller increase in the speed than upon another curve connecting A and C, [p. 364] if the body begins the descent from A with the same speed. Moreover, by this reason it is possible to pick out from the curves found that curve produced, upon which the smallest increase in the speed is generated, or upon which the body is carried with the maximum uniform speed. And in this sense it is easily observed that the motion cannot begin from rest. Though it is certainly the case that if the points A and C have been placed on a vertical straight line, then upon this line from the vertical motion made from rest at A, the maximum speed is produced at C; yet the calculation does not give this solution, even if the vertical line is present, if the initial speed at A is made to correspond to the height k m g , which speed is of such a size, that no further increase can be taken. Therefore with this speed the body descends uniformly from A to C; and for this reason zero is the minimum increase taken in the speed. Therefore the problem, in order that it has an agreeable solution, must be proposed thus : among all the lines joining the points A and C to determine that line, upon which the motion of the body takes the smallest increase in the speed, and likewise to define a fitting initial speed of the body at A.

[Euler finds that the method fails under certain boundary conditions. Thus, if the initial speed is zero, that value cannot be treated as a variable; again, if the body has reached its terminal velocity at A, then the method fails, as no further increase in speed is possible.]

Scholium 3

672. Now problems of this kind ought to follow in a prescribed order, in which suitable curves are to be investigated by a certain given relation of the times ; but since many relations of the times can be reduced to relations of the speeds, I do not bring forwards questions of this kind [p. 365]. However, in this work I will examine the question of the brachistochrone curve, since that, even if the condition of the time has to be prescribed, cannot be reduced to ratios of the speeds, and to which we will now attend. Whereby I will use the same premises as in the above treatment of the brachistochrone in vacuo (361 – 366).