Proposition 83

PROPOSITION 83. Problem

751. According to the hypothesis of a uniform force acting downwards and a uniform medium that resists in the square ratio of the speeds, for the given curve MA (Fig. 84) to find the other curve AN of this kind joined to that at A, in order that the body descending on some arc MA on the given curve by ascending on the curve sought completes the arc AN which is equal to the arc MA.

Solution

On putting the force acting g, with the exponent of the resistance k and with the speed at A, that it has acquired from the descent on the given curve, on each sought curve the ascent AN begins, corresponding to the height b, for the given curve let the abscissa AP = x, the arc AM = s ; for the sought curve now let the abscissa AQ = t and the arc AN = r. With these in place the height corresponding to the speed of the descending body at M is equal to : and the height corresponding to the speed of the ascending body at N is equal to : Therefore the descent of the whole arc comes about from this equation : now the whole arc of the ascending body comes from this equation

Therefore between the arc of the descent and the arc of the ascent, we have this equation : or the differential of this :

But since the curve MA is given, the equation is given between s and x; from which, if in place of dx the value of this replaced by s and ds, the equation is produced between t and s or between t and r since r = s; which determines the nature of the curve sought AN. Q.E.I.

Corollary 1

752. If the lower part of the given curve MA is expressed by the equation x = αs n , then for the lower part of the curve AN there is this equation : dt = αne k s n −1ds ,

Because the smallest arc s is given by : …

thus it becomes :

or only t = αs n . Therefore the smallest parts of each curve are similar to each other.

Corollary 2

753. From the equation e dx = dt it is understood that always dt < dx or t < x. Therefore the point N is always put lower than the corresponding point M. From which it follows that the curve AN bends less towards AB than the curve AM.

Corollary 3

754. On this account the curve ANC cannot be similar and equal to the curve AM, since in this case the points M and N are equal to the arcs AM and AN with the ends to be placed at the same height.

Corollary 4

755. If the body on the curve MA descends from an infinite height, since at A it has acquired only a finite speed, then it is able to ascend to a finite altitude only. Therefore in this case, since the curve AM is extended to infinity, the curve ANC is unable to ascend beyond a certain height, or has a horizontal asymptote BC. Is also apparent from this that if one puts s =∝ ; then indeed there is produced dt = 0.

Scholium 1

756. As it is understood from this proposition, how the curve of ascent AN can be found from the given curve of descent MA, thus hence in turn it is easy from the given curve AN to define the other. If indeed the equation is given between t and s, then …

Corollary 5

757. Since the curve of the ascents AN corresponds to the curve of the descents MA, the equation of this is : ..

thus, if this curve AN is taken for the curve of the descents, the corresponding abscissa of the curve of the ascents is equal to

Corollary 6

758. If in this manner further corresponding curves are sought, then the following series of equations : Abscissa of the curve corresponding to the arc s

Corollary 7

759. Therefore two contiguous curves of this series have this property, that for these connected at the lowest point A the body descending on the first curve ascends on the other an arc equal to the descending arc. Moreover, the smallest of this series of curves s vanishes, and thus is the is the horizontal straight line, on putting n =∝ , since e −∝ k abscissa of the curve itself.

Example 1

760. Let the given line be the right vertical line ; then we have x = s. Hence ANE is sought for the curve of the ascension (Fig.

85. on assuming AQ = t and AN = s this equation is obtained :

With the exponential quantity eliminated, then the equation is : From which equation it is evident that the curve ANE is a tractrix arising from a string of length 12 k upon the asymptote to the horizontal BD. Whereby the height AB of the asymptote is then equal to 12 k . Now if this tractrix ANE is itself taken for the curve of descent then to that there corresponds the curve sought, the abscissa of this ∫ −4 s corresponding to the arc s is equal to e k ds ; [p. 418] which curve therefore is again a tractrix having a horizontal asymptote, of which the asymptote has been raised by the interval 14 k above A, to which the length of the string is equal. Now all the curves of the above series are tractrices which are generated by strings of which the lengths constitute this series : k0 , k2 , k4 , k6 etc. Clearly the vertical straight line is to be considered as the tractrix of which the generating string is k0 or infinite. Moreover the last of this series of tractrices becomes the horizontal straight line drawn through A.

Example 2

761. If the line of the descents is a straight line MA inclined in some manner to the horizontal (Fig. 85), thus in order that MA (s): AP (x) = α :1 or dx = ds , this equation α is obtained for the curve sought AN : −2 s αdt = e k ds , and the integral of this is :

From which equations for the adjoining curve arises

Which is also the equation for generating the above tractrix, with a string of length k2 with the horizontal asymptote BD , on taking AB = 2kα ; and this tractrix must pass through A. The following curves of the series are all tractrices also, as in the preceding example, of which the generating strings are k0 , k2 , k4 , k6 etc. , now of these asymptotes the distances from the point A are held in this progression [p. 419] 0kα , 2kα , 4kα , 6kα etc. Clearly all these tractrices make an angle with the vertical axis AB equal to the angle PAM.

Corollary 8

762. Of these tractrices that, which first or precedes the line MA, hence has this property that the body descending on that afterwards ascending on the line AM traverses equal distances.

Corollary 9

763. Therefore according to the curve of the descents CA (Fig. 86) to be found, to that corresponds the right inclined line AM, above the asymptote to the horizontal, the tractrix CA is described by a string of length k2 and on that the applied line Ab = 2kα is taken, and from A the right inclined line AM is constructed ; and then CA is the curve of the descents, to which there corresponds the line AM for the ascents.

Scholium 2

764. Here the counterpart case can be of service, examples of the inverse problem, in which from a given curve of the ascents the curve of the descents is required.

Example 3

765. Let the curve of the descents be the given cycloid MA (Fig. 84), the nature of which can be expressed by the equation

2ax = s 2 or the diameter of the generating circle is equal to a2 . Hence it follows that dx = sds , a and thus the equation is found for the other curve of the ascents AN −2 s adt = e k sds , the integral of this is

which on account of

goes into this equation : This curve at A, as has now been said, has a horizontal tangent. Now also it has a horizontal asymptote BC; the height BA of this is found if s becomes equal to ∝ .

Moreover in this case making e k = 0 ; whereby it then follows that t = AB = 4k a . From this it is understood that the curve must have a point of inflection somewhere ; which can be found, if on putting dt constant there is placed dds = 0. Hence now there is produced 1 = 2ks or s = k2 . Whereby if the arc is taken AN = k2 , then N is the point of

inflection; to which there corresponds the abscissa AQ = 4k a − 4kae or BQ = 2kae . Concerning which is always the case that AB : BQ = e : 2 = 2,71828 : 2.

Scholium 3

766. This problem was proposed anonymously in the Act. Lips. A. 1728 [Problema geometris propositum, p. 528.] and the solution of this was given in the Comment. Acad. Petrop.A. 1729 by the most distinguished D. Bernoulli who used another method. [Dan. Bernoulli, A theorem concerning the curvilinear motion of bodies that experience resistance in proportion to the squares of their speed, Comment. Acad. sc. Petrop.4 (1730/1), 1735, p. 136.] [p. 421] Now beside this condition, that anonymous person required chiefly a single continuous curve, one branch of which was for the descent, and the other serving for the ascent, and innumerable curves of this kind are given, which we uncover in the following proposition.