The Motion Of A Point On A Given Line In A Vacuum

PROPOSITION 14. Problem.

106. If there is an infinite family of similar curves AM, AM, etc. (Fig.17) beginning from the fixed starting point A, to find a curve CMM from these other curves cutting the arcs AM, AM, etc, which are traversed in equal times by a body descending along these arcs, as before, with a uniform force present acting downwards everywhere.

Solution

From the infinite number of curves given one is taken AM, the parameter of which is a. Putting in place AP = x, PM = y and with the arc AM = s and as before the force acting being equal to g , the body descends on the given curve AM; the speed at M corresponds to the height gx.

Hence the descent time on AM is equal to ds . Hence … from all these curves AM, AM, etc., so many arcs are to be cut, in order that from these the quantity ds is constant.

But … ∫ dsgx is referring to other curves, if besides s and x, the parameter a also is made a variable.

Therefore with a made variable as well in ds , the quantity ds is indeed constant for that time in … which all the descents are to become the same. Let this time be equal to k, and k = ds for individual curves.

Whereby if ds is thus differentiated, as also a is placed

…

variable, this differential is put equal to zero. In order that the differential of this integral can be found, let ds = pdx and p is a function in which a and x likewise make numbers with zero dimensions [such as a function of x/a]. [p. 49] Hence we have the integral .. ∫ gx ; this differential with the variable a in place also, gives

…

which must be equal to 0. Now the quantity q is to be found in the following way. Since k= ∫ gx , in the quantity k [or in dk], the variables a and x produce a number having the pdx dimensions 12 . Moreover this is shown elsewhere, in Vol. VII Comment.

However, if u is a function of m dimensions of a and x, and du = Rdx + Sda; then um x is a function of a and x zero dimension. Therefore um is differentiated, and there is …

produced . Because it arises from the differentiation of a or

…

function of zero dimensions, Rx − mux + Sax = 0 , or Rx + Sa = mu . Whereby if u is a

function of dimension m of a and x themselves; and the equation du = Rdx + Sda is put in place, then Rx + Sa = mu and thus du = Rdx + da

( mu − Rx ) or a adu = Radx − Rxda + muda . '

Thus, in the present case, R = p , S = q, m = 12 , and gx u = k .

In addition, the establishment of functions of zero dimensions is discussed in E012 in these translations.], then the equation arises From which it is found that [there is a misprint present here in the O. O., but not in the original text.]

Therefore we have which is the equation for the curve sought. But if the equation between the coordinates x and y for the curve CMM is desired, from the equation for each of the curves AM, the value of a found must be substituted into the equation between x and y. Q.E.I.

Corollary 1

107. Also the equation found at first, is sufficient for the curve CMM to be found. As for any given abscissa AP = x , from that the parameter a of this curve AM is found, of which the corresponding point M of the assumed abscissa x lies on the curve CMM sought.

Corollary 2

108. Moreover since this is a differential equation, and thus to which more curves pertain according to the added constant, it is to be noted that with the addition of the constants, only that solution is to be agreed upon for which the given curve is completed in the time of descent k, for the given value of a that gives the abscissa x only of the required arc AM to be cut off.

Corollary 3

109. If the time of descent k must be equal to the time of descent through the vertical distance AC = b, then k = 2 b . With which value put in place, we have the equation :

In the integration of this equation, it has to be arranged that the curve passes through the point C.

Scholium 1

110. Moreover it is always the case that the vertical line AC arises as a kind of curve AM, if the parameter a is taken to be indefinitely large or small. Whereby the constant time k is most conveniently expressed by the descent through the vertical AC, clearly a kind of curve AM. And in the construction of the equation found a constant of such a size is to be added, as by putting x = b, a is made infinite or zero, as one or the other value corresponds to the value of the line AC.

Scholium 2

111. If it is possible to integrate ds itself, without the aid of any equation, by means of …

which q can be found. For if the integral ds is itself again differentiated with respect so

…

some variable a also, q is again found; and it is only necessary to put this differential equal to zero. Now most conveniently in these cases the problem is solved, if the integral of ds is at once put equal to k or to 2 b and in place of a the value is substituted in …

terms of x and y from the given equation for the curve. [p. 51] And in this way the solution is established not only for similar curves, but also for dissimilar ones, but only if the descent times can be expressed by a finite quantity.

Exemplum 1.

112. If all these curves AM are straight lines inclined in different ways to the vertical AC, then where n is to be considered as a parameter. Hence the time becomes which must be placed equal to 2 b itself. Thus there becomes g y Moreover since n is a variable quantity, put the value x for that from the equation y = nx; with which accomplished, this equation is produced for the curve CMM between the orthogonal coordinates x and y : which is the equation for the circle, the diameter of which is the line AC = b.

Scholium 3

113. This case is the one examined before (102); indeed there it was shown that the body descends in equal intervals of time by all the chords drawn in the circle from the uppermost point. Here indeed the case does not concern similar curves, but we report on this example as a case illustrating scholium 2, because for straight lines and for these, the descent times are expressed by finite quantities. Now the following examples will include similar curves, as the proposition postulates. [p. 52]

Example 2.

114. Let all the curves AM, AM be circles tangent to the vertical AC at A. The radius of each of these is equal to a, and it is given by

Now these circles are all similar curves, because a, y and x in the equation keep a number of the same dimension, or they complete the homogeneity alone. Therefore the radius a must be handled as a variable parameter. Moreover, there is obtained from that equation

…

, whereby we have p = .. and thus it has the prescribed property,

…

as the dimension of the number formed from a and x is zero. On this account we have this equation for the curve CMM : or this

Which equation can be solved; for on putting x = au there is produced in which the indeterminates are separated from each other. Moreover, in which the equation is obtained for the curve CMM between the coordinates x and y, in place of a is put the value

..

and in place of the differential of this … With

…

which put in place the following differential equation is obtained Which thus must be integrated, so that with x = b it makes y = 0, which curve must pass through the point C.

Corollary 4

115. From this equation the tangent of the curve CMM is known at individual points and from the position of the tangent the angle AMM is known, [p. 53] in which whatever of the given curves is divided. Clearly the tangent of the angle AMM =

…

Therefore .. here the angle at C is right on account of x = b, or the curve CMM is normal at C to AC.

Corollary 5

116. If b is taken either greater or less, the curve CMM is different also and in this way an infinity of isochronous curves arise being cut from the circular arc. And these curves are all similar between themselves on account of the parameter b, which constitute a homogeneous equation with x and y. Hence with one given curve CMM innumerable others can be constructed from that, clearly with the x and y coordinates of the curve CMM augmented or diminished in the same ratio as AC or b is augmented or diminished.

Example 3.

117. Let all the curves AM, AM be cycloids having cusps at A and vertical tangents AC at A. With the parameter of any cycloid AM put in place, or with twice the diameter of the generating circle equal to a, from the nature of the cycloid, we have s = a − ( a 2 − 2ax ) and ds =

…

hence dy = dx2 2ax . Hence in this case we have the

( a − 2 ax )

function of a and x of zero dimension,

…

required. Whereby for the curve CMM, this equation is required

If the equation between the orthogonal coordinates x and y is desired, from the equation or with the differential of this likewise, for the variable a, the value of a itself must be substituted.

Now this differential equation with the variable q in place gives or Which goes into this : Now the above multiplies by

…

provides this

These two added equations give an integrable equation, the integral of which is : From which the value of a elicited becomes : and

With which values in the equation : which arises from the two differential equations with da eliminated, on substitution gives the equation for the curve sought CMM.

Corollary 6

118. From this equation the tangent of the angle is found that the curve CM makes with the applied line PM, truly

Then also the tangent of the angle becomes known, [p. 55] that the cycloid AM makes with the applied line PM. From the equation of the cycloid is without doubt

Now on eliminating a the tangent is equal to …

Whereby, since either of these given angles is the complement of the other, hence with this taken into account, the angle that the curve CMM makes with any of the given AM is right. Consequently the curve CMM is the orthogonal trajectory of all the given cycloids AM, AM, etc.

Corollary 7

119. With AC taken of another size also other curves CMM are produced and thus an infinity of orthogonal trajectories are found, which are all similar to each other. Hence from one easy given, it is possible to construct any number you please.

Scholium 4

120. All these isochronous curves being cut from the arc, whatever the curves cut, can always be constructed, even if it is not apparent from the equation. For by quadrature the arcs which are completed in a given time of descent can be removed from the given curves, and in this way any points on the curve sought can be found.

If certain curves cut are algebraic, then the equation for the curve cut can always thus be compared, as by making a substitution of the indeterminate, which can then be separated in turn from each other.

But if the curves cut are expressed by a differential equation, [p. 56] the differential equation for the curve cut most rarely admits to being separable in terms of the indeterminates.

Because, in a singular manner, as I have used in the case of the cycloid, the parameter a can be eliminated and there the substitution cannot deduce a separation.

Scholium 5

121. Then it is necessary to observe that all the isochronous curves cut by an arc, the number of which is infinite, are similar to each other, according to the value of the variable b, if indeed the curves cut are of such a kind.

This is gathered from the general equation in which, since p is a function of zero dimensions of a and x, and the quantities a, b and x constitute a homogeneous equation.

But from the equation of the curve cut, since in that equation a, x, and y are put in place everywhere to make a number of the same dimensions, the value of a is a function of x and y of one dimension. Whereby with that substituted in place of a, the equation is obtained for the curve cut, in which b, x, and y everywhere constitute a number of the same dimensions. Consequently, for the variable b put in place, there arises an infinite number of curves similar to each other with respect to the point A. Hence with a single curve given, the rest can easily be described by reason of the similitudes.

Scholium 6

122. Now this material concerned with the cutting of isochronous arcs was published in the past [p. 57] in the Act. Erud. Lips. A. 1697 [p.206] [The radius of curvature in translucent media ….and concerning synchronous curves, or the construction of rays from waves; Opera Omnia, Book I, p. 187] by the Celebrated Johan. Bernoulli and later in the Comment. Acad. Paris by the Cel. Saurino, [1709, p. 257 and 1710, p. 208 ; General Solution of the problem, …..] who indeed used another method. Now I have used that method that I have treated in our Comment. pro A. 1734, as the most convenient for the solving of this kind of problems. Now in their works these celebrated men only considered similar curves as I have done, since without doubt for dissimilar curves the solution can be exceedingly difficult and often also too hard to solve. Now these curves are called synchronous in the places they are cited, since arcs traversed in the same time are cut off.

[Note that there was a delay of several years between the writing and the eventual publishing of Euler’s works, even at the start of his career at St. Petersburg; thus occasionally he was able to extend his observations from volumes to be published later into earlier ones that had not yet been published either, as below. Thus, one must take the Enestrom Index with a pinch of salt as regards the chronological order of the works, as there was some coming and going, and of course a number of papers did not make it into the index at all in the original assessment, which should be looked at by somebody with an interest. The original Euler Archive is of course at St. Petersburg, and access does not seem to be that simple, and neither is it free.]

Scholium 7

123. It is apparent, as shown from my dissertation in Vol.VII Comment. Acad. Petrop. that these synchronous curves can be found in a like manner, also if the given curves are not similar, but yet of such a kind that, as with ds = pdx in place, in p the quantities a and x constitute a number of given dimensions; for then it is equally easy to find the value of the letter q ; as if the number of the dimensions of a and x in p is n, this equation for the curve is found : Whereby if n = − 12 , so that we can make p = ac , then we have dx …

and thus …

x = ma, or x can be taken in a given ratio to the parameter a ; therefore in which case the construction of the isochrones is most easy. [p. 58] But if p does not have a value of this kind, from my paper cited above it is understood how the required equation has to be sought.