Proposition 30

PROPOSITION 30. Problem

270. Under the hypothesis of a force acting uniformly and tending downwards, to find the curve AM (Fig. 35), upon which a body descends uniformly receding from a given point Ct.

Solution

Let AM be the curve sought; the tangent CA is taken, which passes through the given point C ; the speed of the body at A is a minimum. Indeed since the total speed at C is devoted to moving away, in other elements of the curve it is necessary, since the speed is greater, that only a part of this is taken for receding. The point A is therefore the highest point of the curve sought.

Therefore let the speed of the body at A correspond to the height b and with this speed the body begins to move uniformly along AP ; and thus this motion with the descent of the body agrees with the motion along the curve AM , so that at any point P and M equally distant from C is reached at the same time. With the speed at M corresponding to the height b and taking CP = CM = x and let the sine of the angle PCM = t with the total sine taken as 1. The circular arcs PM and pm are drawn with centre C; then Mn = Pp = dx and the sine of the angle pCm is equal to t + dt. Whereby we have :

Therefore we have .. v must be described in the same time as the ..

Hence since the element Mm with the speed element Pp with the speed b , it becomes : or Therefore it is required that v is determined. In order to do this, the vertical CQ is drawn from C and the horizontals AD and MQ; therefore after the body as descended from A to M, it has fallen by the interval DQ. Whereby with the force acting put as g, then we have

Let AC = a, the sine of the angle ACD = m; and the cosine is equal to ( 1 − m 2 ) , thus CD = a ( 1 − m 2 ) and

On this account :

From which is constructed : In which with the value v substituted there is produced this equation : or this :

Which equation expresses the nature of the curve sought, and if the indeterminates x and t can be separated from each other in turn, then the curve can be constructed. Q.E.I.

Corollary 1

271. Therefore it is evident from the equation found that there are innumerable curves to satisfy the question, on account of the three quantities : clearly the angle ACD, the distance AC and the speed b , from which the body recedes from the fixed point C, which can be varied as it pleases.

Corollary 2

272. And of these three quantities, any two can be assumed arbitrarily and from the third variable only an infinite number of curves are produced satisfying the question. But since this equation cannot be constructed generally, all the satisfying curves cannot be shown.

Corollary 3

273. Because it restrains the figure of these curves, it is understood that all these must have the same cusp at A, because A is the highest point. Otherwise indeed a branch of the curve from A must descend to another part of the line AP, with the exceptional case in which CAP becomes a horizontal line ; for then this argument comes to an end.

Corollary 4

274. Now another branch put equal to another part of the line CP solves the problem and gives rise to that AM. Indeed it is found from the same equation, but if t or the angle PCM is taken negative.

Corollary 5

275. But from the single equation found, by inspection it is evident that for two cases the indeterminates can be separated, of which the one is, if a = 0, and the other if m = 1. Clearly in that case the distance AC vanishes and the point A is incident on C; now in this case the straight line CP is made horizontal. [p. 123] We explain both these cases in the following two examples.

Example 1

276. Hence the point A is incident on C, or the descending body begins from the point C itself; making a = 0. Therefore in this case the equation for the curve sought changes to this : in which the indeterminates are separated from each other. Therefore the construction of the curve sought can be made by quadrature ; indeed it becomes : which integration can thus be completed, in order that on making t = 0 , then x = 0. And for the general equation thus to be integrated, as by putting t = 0 makes x = a. Therefore in this case the integral

…

thus can be taken so that it vanishes by making t = 0. Now in the construction of this integral it is observed to be better if I put the cosine of the angle MCQ or with which done it becomes the sine of the angle MCm or With these substituted this equation is obtained : which integral can thus be accepted, as on making q = ( 1 − m 2 ) it becomes x = 0.

Corollary 6

277. If different values of b are given, all the curves which arise are similar to each other ; [p. 124] for with the angle MCP maintained, the proportional distance CM is taken for b, the height generating the initial speed.

Corollary 7

278. Therefore whatever the angle ACQ may be, the construction is not changed, but only a constant is to be added. Whereby the construction serving one case can accommodate all the cases.

Scholium 1

279. This problem concerned with uniform recession from a fixed point previously was proposed and solved in the Act. Lips. A. 1694 and the solutions presented there agree extremely well with the case of this example; indeed the general solution was not given in that place. On account of which the case of the following example is clearly seen to give anew curves satisfying this equation. But since the following construction agrees with that, though the curves are clearly different, yet also the following case for these, which are treated here concerning this, is considered to be contained. Moreover, curves of this kind are called paracentric isochrones, since the motion upon these is uniform from a fixed centre.

Example 2

280. Let the line CAP be horizontal; put m = 1 and the term ga ( 1 − m 2 ) vanishes in the general equation. [p. 125] Therefore in this case the equation becomes separable as before ; for the general equation is transformed into this : which integral can thus be taken, as on placing t = 0 it becomes x = a. Whereby dt thus on integration, as it vanishes with t = 0, hence : 3 ∫ ( t −t ) Which construction hence agrees with the preceding.

Scholium 2

281. Whether besides these two cases others are able to be found, that admit separation of the indeterminates, I doubt very much. Certainly no one, as far as I know, has elicited another, on account of which I judge that it is not necessary to tarry longer over this material.