Chapter 6n

Proposition 121

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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Table of Contents

PROPOSITION 121. PROBLEM

  1. If the centripetal force varies as some power of the distances from the centre, and the body moves in a medium with constant resistance, which resists in the square ratio of the speed, to determine the curve AM (Fig.91) that the body describes, and the motion of the body on this curve.

SOLUTION

By keeping as before: CM = y, CT = p, Mm = ds, with the speed at M corresponding to v there becomes :

Hence this equation is found for the curve sought :

However not a lot can be understood from that equation about the curve put in place, on s account of the complicating factor e c ; whereby with the logarithms taken, there is and on taking dy constant. Since indeed ds = ydy ( y2 − p2 ) , we have

…

Which is the equation between y and p sought for the curve. Q.E.I.

Corollary 1

  1. From which an equation is produced, if the force is proportional to the distance or inversely proportional to the square of the distance, that is readily apparent from the equation found, if 1 or – 2 is substituted in place of n. Moreover all the substitutions of this kind do not help in bringing about the tractability of the general equation.

Corollary 2

  1. If the medium put in place is not uniform, but the exponent of this is the variable ds q, in place of e c there is put e ∫ q (873) [p.434] and this equation is found for the curve described : s Where, if q is made proportional to the distance y, the equation can be reduced to a differential equation of the first degree.

Corollary 2

  1. Therefore let the exponent of the resistance q = α , and the curve described can be expressed by the following equation :

In which, since the number of dimensions of the individual terms vanishes, reduction to a differential equation of the first degree can be put in place.

Corollary 3

  1. Moreover in this manner the differential equation of the first degree is found.

Putting there is − sdt 2 . Again we have Whereby the equation becomes ddt = − dzdt z and From which we find

Corollary 5

  1. If the centripetal force is put to vary inversely as the cube of the distance, then n = −3 . [p.435] Hence the curve described is contained in this equation :

Corollary 6

  1. If the centripetal force varies in proportion to the inverse square of the distance, then n = −2 . And the curve described is expressed by the following equation : In the same manner, if the centripetal force is proportional to the distances or n = 1 to be put in place, there is produced :

Corollary 7

  1. All these equations give curves in the vacuum if α = 0 is put in place. For in this case the exponent of the resistance is made infinitely great, and therefore the resistance infinitely small. Moreover, this equation is found :

Scholium

  1. Therefore when the exponent of the resisting medium, which is put to resist as the square of the ratio of the speeds, is proportional to the distances from the centre, then the equation for the curve described can be reduced to a differential equation of the first degree ; which is hardly possible to be done for the other hypotheses of the exponents of the resistance. Moreover I understand that such values of q, which only depend on the distances from the centre y, are clearly the only ones admitted to be put as a ratio. [p.436] Indeed it is not fitting to give q in terms of p, i. e. through the curve itself, which is still unknown. Yet meanwhile the differentio-differential equation can always be reduced to a first order equation [note that Euler considers differential equations as those written with single derivatives such as dx, dy, etc; while he considers the ratio of differentials such as e. g. dy/dx to be the ratio of a differential by a differential], as long as q is given as a function of a single dimension of y and p taken together. But with these equations, even if they are differentials of the first order, they are neither able to be separated nor integrated, and they serve no useful purpose. On this account, we consider the resistance proportional to the speed when it is joined with a centripetal force to any power of the distances.

[Thus, there are now three kinds of variables to be considered : the power law governing the central attraction; the relation of the resistance to the speed; and the form of the function formed from the exponent of the resistance; only a small portion of these cases is soluble.]

PROPOSITION 122. PROBLEM

  1. In a uniform medium, which resists in the simple ratio of the speed, the body moves attracted to the centre C (Fig.91) by a force proportional to some power of the distance ; the determine the curve AM that the body describes.

SOLUTION

By putting CM = y, CT = p, Mm = ds, with the speed at M corresponding to the height v and with the exponent of the resistance equal to q, the centripetal force is equal to yn fn and the ∫ area ACM = 12 pds = S . With these put in place we have (1012 and 1014), [p.437] where b is the height at A corresponding to the speed, and h is the perpendicular from C sent to the tangent at A. From which S can be eliminated, and the equation found put in this form : From which by differentiation with dp put constant there arises : or

…

Which equation expresses the kind of the curve AM described. Truly by knowing this, at once the speed of the body can become known from the area of the curves and from the perpendicular p. Q.E.I.

Corollary 1

  1. If in place of dp the element dy is assumed constant, then this equation is produced: Moreover nothing can be concluded from this equation, since it cannot be reduced to a first order differential equation.

Corollary 2

  1. The above reduction can always be put in place (1020), if in the differential equation the indeterminates [p.438] p and y are agreed to have the same number of dimensions. Moreover, this happens if n = 1, i. e. if the centripetal force is proportional to the distance from the centre. Then indeed for the curve sought : with dy put constant.

Corollary 3

  1. Hence therefore for this hypothesis, put : and there is produced : With these substituted it is found that Or by putting tz = u there is produced:

Corollary 4

  1. This equation is possible to be integrated, if it is divided by tt( 1 + u ) 2 u ; from which there is produced :

The integral of which is or

Corollary 5

  1. Truly on the strength of the substitutions made: On account of which we have the equation for the curve sought:

Corollary 6

  1. Moreover when the differentials are made rational, there arises : by restoring p = yt. Therefore the following equation is found, in which the indeterminates y and t are in turn separated from each other, From which the equation for the curve can be constructed.

Scholium

  1. I will not delay any more over this equation, although I suspect that it is possible to be integrated anew. This indeed is certain, if C 2c = − f ; in which case as the integral is made composite, as here I did not wish to make the change. From which it is understood that the integral obtained is very complicated, so thus hardly anything could be deduced about understanding the motion. On which account with these in place I go on to the inverse problems.

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