Proposition 88

PROPOSITION 88. Problem

799. If the curves MA and AN (Fig. 87) have that property, that all the semi– oscillations which begin on the curve MA, are between themselves isochrones in a medium that resists in the square ratio of the speeds, to determine the case in which these two curves MA and AN joined together constitute one continuous curve.

Solution

With the same denominations put in place as in the previous proposition, clearly AP = x, AM = s, AQ = t, AN = r , and f is

equal to the length of the isochronous pendulum in vacuo and with gravity equal to g, there we found these two equations :..

in which the relation between each curve is contained. Now since the curves MA and NA must be the roots of a continuous curve, the equation between x and s thus must be compared so that, if x changes to t, then s becomes equal to – r on account of the negative in place. According to this we accept the new variable z, and both s and x are such functions of this, that on making z negative, x changes to t and s to –r . Let [p. 443]

then [above] we have for on making z negative, in which case r is changed into –s and –s into r, then there is produced : which equation agrees with the former. Let P be some even function of z, which is not changed, even if –z is put in place of z, and there is put in place

…

with which put in place the question is satisfied. And if we make z negative; then –s changes into r and there is obtained : as required. For the other equation is now satisfied by this

for on putting z negative and dt in place of dx and r in place of –s , there is produced : Therefore from the variable z, of which P is some even function, the curve sought AM of which the continuation is the other AN, is thus determined, in order that it becomes

Hence it follows that

Let z be become

which enables simpler formulas to be put into effect as well as producing more convenient homogeneous equations, as u must be of a single dimension; thus P also is an even function of u of one dimension. Whereby there is obtained : Q.E.I

Corollary 1

800. Hence an infinitude of tautochronous curves MAN can be found, if from the different values of P that can be put in place, all those substituted are even functions of u. Now the equation between x and s is obtained, if from the two equations found the variable u, which also is not present in P, can be eliminated.

Corollary 2

801. Since we have s = 2kl Pk−u , then

… and … and

Since with which equation, …

if it can be combined with the other, there is produced Which equation is often the most convenient in the elimination of u.

Scholium 1

802. Since with s vanishing, x also must vanish, the first to be investigated is that in which u itself vanishes for a given value of s. Then the integral thus must be taken, in order that it vanishes, if the same value is substituted in place of u. And this is to be observed since in the construction of the curve, which can be put in place with the help of the two equations found, then in putting together the equation between x and s, if indeed from the equation integrated ..

this can be deduced. Otherwise if in place of u and P some other multiple of these can be put to use, in place of the two equations found these can be used : where the constant c is arbitrary and thus in this manner can be determined, as in the same case that s vanishes, in which x vanishes. But s vanishes if u = 0, since

…

and e dx vanishes on s vanishing ; whereby c must be equal to the value of P, if u is put equal to 0 in that. Therefore in the same case x must vanish, from which the constant in the integration of the value of x is determined. Or even in place of P such an even function of u must be accepted, which becomes equal to c, if u is put equal to 0.

Corollary 3

803. Since the length of the isochronous pendulum in vacuo and with gravity equal to g is equal to f and the smallest oscillations in a medium with resistance are in agreement with the oscillations in vacuo, the radius of osculation of the curve at A = f, if indeed the tangent to the curve at A is horizontal.

Example 1

804. Because P must be an even function of u, let P be the constant c = k, from which on putting u = 0 , making s = 0. Hence it follows that And on account of dP = 0 there is obtained [p. 446] From which equations there is thus put in place …

Which equation is for the tautochrone of the descents ; which made continuous beyond A gives the curve of the ascents ; and all the successive semi–oscillations upon this curve, provided they start from MA, are isochronous.

Example 2

805. Let then P keeps the same value on making u negative. With this in place, then then and on account of dP = 2udu a from which equation there is produced :

Which value of u substituted into the other equation gives : and with the root extracted :

In the special case, if a = 4k, this equation becomes : in which two equations are contained, of which the one is : and the other : …

Moreover the latter of these, since dx does not vanish on putting s = 0 and on account of negative dx is useless. Now the first equation gives : or Which made continuous beyond A is expressed by this equation : Now through a series this [first] equation is obtained : and for the other part AN this series :

Corollary 4

806. Since … then ..

Now on eliminating u there is obtained : Whereby if that value of u is substituted into this equation, then the equation between s and x is produced at once.

Example 3

807. We may put in place substituted in place of P and u2 with the values given above, then Hence with the square taken, there arises : Now this equation differentiated gives this : [p. 448] the integral of which is :

Which equation converted into a series gives :

Scholium 2

808. Which tautochronous curves we have found in these examples for a medium that resists in the ratio of the square of the speeds, these have thus been composed so that the arcs MA and AN are dissimilar. Therefore since all the descents must begin on the curve MA, the following semi–oscillations, which begin on the curve NA, are not tautochrones, and because of that these curves cannot be adapted for oscillatory motion. But a remedy for this inconvenience is produced, if of the curves of this kind, even similar and equal curves MA and NA are found ; for in this case descents are likewise able to be made on each curve. Also there is no doubt that such a case exists, and the discovery of this, since the two curves perhaps are not continued, pertain rather to the preceding proposition. Clearly the curve of the descents must be investigated, to which there corresponds a similar and equal curve of the ascents ; now this investigation is thus difficult on account of deficiencies in the analysis, as I doubt that the goal can be reached without the development of some outstanding analysis.

This question is thus reduced so that the equation between s and x of this condition can be investigated, as, if in that equation is put in place of s and in place of x, the same equation can be produced, which was had before [788]. Indeed this condition can be more easily effected in many ways; yet I cannot see how that condition can be satisfied. If the medium should be the rarest, then it is not difficult from what has been reported on, to find the case in which two curves MA et AN are similar and equal to each other. Indeed in the end by leading through this calculation I have found the equation which curve likewise on being continued beyond A has the branch AN similar and equal to the arc AM; whereby a pendulum oscillating on this curve completes single semi–oscillations in equal times. Moreover we have : Because now k is a very large quantity, then Hence the equation becomes :

On putting f = 2a; then

Which curve can hence be described in almost the same manner in which the cycloid is described with the help of the rectification of the circle.

Corollary 5

809. If there is taken a = k 3 or f = 2k 3 , this curve changes into an ellipse, the horizontal axis of which is twice as great as the vertical axis, which is equal to 2k 3 . Hence it can happen, that an ellipse can be the tautochrone in the rarest medium and is more satisfactory than the cycloid.

Scholium 3

810. Moreover the construction of the tautochrone curve in the rarest medium in the preceding scholium has been given as follows: On the vertical straight line AB = a = 12 f (Fig. 88) the semicircle AOB is described, and from this on the base BD the cycloid AFD is described; with the same inverted in place AGD is described. With which accomplished the curve sought AMC is constructed by taking everywhere for the applied lines of this : from which ratio endless points of the curve become known. Or it is possible to take thus so that there is no need for the cycloid. Moreover this curve has a vertical tangent somewhere or the applied line PM is maximum, which is found on putting dy = 0.

Moreover there is produced : if AP is taken equal to this value, then the maximum applied line is found.