Proposition 86

Table of Contents

PROPOSITION 89. Problem

According to the hypothesis of gravity acting uniformly downwards g with some given curve am (Fig. 89) for the descents in vacuo to find the curve AM for the descents in a medium of this innate character with uniform resistance in the ratio of the square of the speeds, in order that all the descents on MA are isochrones with respect to all the descents on ma, if the speeds at the bottom points a and A are equal.

Solution

Let the abscissa for the curve am of the descents in vacuo abscissa be ap = t, the arc am = r; now the exponent of the resistance is put equal to k ; now for the curve of the descents in the medium with resistance let AP = x and AM = s; Now two descents are considered on these curves, in which the speeds acquired at A and a are equal and correspond to the height b. Hence the time of the descent in vacuo is equal to and after integration there is put in place gt = b. But for the time of the descent on the curve MA in the medium with resistance there is obtained : if likewise after the integration there is put in place : On account of which [on comparing the integrals] these times are equal, if for with these put in place for each time this expression is obtained …

Therefore since we have ds = dr , then on integration it becomes :

thus there is produced [p. 452] ∫ −s Now the other equation e k dx = t gives −s e k dx = dt

Moreover then we have … in which with the value substituted, there is found … from which there arises :

Hence from the given equation between t and r for the curve am with the help of these two equations, from which s and x are determined by t and r, the curve sought AM can be constructed. Now the equation between x et s can be conveniently found from the given equation between t and r, if in that in place of r there is substituted .. Q.E.I.

Corollary 1

Around the lowest point a, where t and r are vanishing quantities, there becomes or Whereby the inclination of to the curve MA to the axis at A is equal to the inclination of the curve ma at a.

Corollary 2

Again the radius of osculation at the lowest point a, if the tangent is horizontal, is and at A, since the tangent also is horizontal, is equal to equal to rdr dt Hence it follows that Whereby as r is infinitely small, then

Corollary 3

If therefore the curve ma has a horizontal tangent at a, then the tangent of the curve MA at A and the radius of osculation at A is equal to the radius of osculation at a. [p. 453]

Corollary 4

Therefore if the curve found ma in vacuo, in which the times of the descents have some relation to the speeds acquired at a, likewise the problems for the resisting medium can be solved for the curve MA, which by the prescribed reason can be constructed from the curve ma.

Corollary 5

Therefore if the curve ma should be a cycloid or tautochrone in vacuo, then AM is the tautochrone of the descents in the resisting medium found above. For on putting r 2 = 2at or rdr = adt this equation is produced on substituting the values found in place of r and t :

Example

Let am be some right line inclined in some manner, thus in order that r = nt ; then the time of the descent, in which a speed is generated corresponding to the height b, is equal to …

Therefore the same property is in place for the curve MA, so that the time of each descent in the resisting medium, in which the speed b is generated, is equal to 2ng b or in proportion to the speed arising. Moreover since r = nt, then dr = ndt; in which if the values found are substituted in place of dr and dt, then there is produced : [p. 454] which is the equation for the tractrix generated by the thread of length 2k, such as is shown in Fig. 86, truly the curve CA, which has the same inclination at A as the right line ma.

Scholium 1

To the extent that this curve MA has been found, on which all the descents in the resisting medium are completed in the same times as the descents in vacuo on the curve ma, if the final speeds at A and a should be equal, thus in the same manner the curve MA can be defined, upon which all the ascents in the resisting medium are completed in the same times in which likewise with the same starting speeds of ascent the motions are completed in vacuo on the curve am. For since for a resisting medium the descent and the ascent can be changed into each other on making k negative, if on putting AP = x and AM = s, there is obtained : or conversely, From which then the curve AM can be constructed easily from the equation to be found for that.

Scholium 2

In this problem we have determined the curve of the descent in the resisting medium from the curve of the descent in vacuo. Moreover it is readily apparent in turn from the given curve AM for the resisting medium the other am for the vacuum can be found.

For since it is given by : the construction of the curve am with the help of the two equations is performed. Now the equation for the curve am between t and r is found more conveniently from the given equation between x and s, if there in place of x there is substituted 4k 2 dt and 2kl 2 k in place of s. Because here besides what has been said about the 2k − r (2k − r ) 2 descents, likewise is valid for the ascents, but only if k is put negative, as we have advised in scholium 1.

Scholium 3

The devising of one curve from the other of two given curves am and AM also has a place and is treated here, if the equation of the given curve is not in place, but if it has been drawn by hand in some manner; for the construction can be deduced from the formulas, since it no longer depends on the equation. On this account in the preceding chapter (432), in the case we have considered for the vacuum, in which the curve cm (Fig. 90) that we have devised must be joined to the given ac, so that all the descents from any point of the curve cm as far as to a are completed in equal times, now similar examples to these for the resisting medium can be elicited, in which the line composed from the two different curves is a tautochrone. For if the curve acm is a tautochrone curve of this kind for the vacuum, from that by the solution of this problem a like curve can be found composed for the resisting medium.

Clearly from ac by the method proposed the curve AC is defined; which is found on putting bp = t, cm = r and BP = x and also CM = s and besides ab = a, ac = c ; AB = A, and AC = C, for then, since the equation between t and r is given, then and [p. 455] But if for the resisting medium the curve AC is given and there is required for the other part of this CM the property, in order that all the descents on MCA are completed in equal times, the solution can be effected in a not dissimilar manner. For from the given curve AC for the resisting medium there is found the curve with the same property for the vacuum ac by scholium 2. With which devised, there is sought the curve cm to be joined to that, which produces all the isochronous descents in vacuo (432).

Then by the method treated in the manner according to the composite curve acm for the vacuum there is sought a like composite curve for the resisting medium ACM, of which indeed the part AC is now known ; clearly from that line ac we have defined. Hence likewise the problem, in which in vacuo there was some difficulty, in the resisting medium too can be resolved. Hence finally the chapter ends and I ask the benevolent Reader, that before progressing to the following chapter, it may be wished to repeat what has been presented in chapter I from § 58 to the end of the chapter.