Chapter 2r

Proposition 31

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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PROPOSITION 31. Problem.

  1. With a uniform force present acting downwards, to find the curve AM (Fig. 36), upon which a body with a given initial speed moves thus, so that in equal times equal angles are completed about a fixed point C.

Solution

The beginning of the curve is taken at a certain place A, at which the line CA is normal to the curve itself, and let the speed at A corresponding to the height b and let AC = a ; the angular speed is as ab , [p. 126] to which quantity the angular speed expressed at individual points M must be equal. Let the speed at M correspond to the height v and CM = x, then we have mn = dx. The ratio is made so that

[The component of the speed normal to the distance CM is taken.] since the quantity divided by MC gives the angular

…

Mn . v ; which since this is equal to ab , we speed equal to Mm .MC have this equation : Now let the line DCQ be drawn vertical and the sine of the angle ACD = m, the cosine of this is equal to ( 1 − m 2 ) with the whole sine put equal to 1. Likewise the sine of the angle MCD is equal to t; and the cosine is equal to ( 1 − tt ) . Therefore with these in place it follows that CD = a ( 1 − m 2 ) and CQ = − x ( 1 − tt )

…

and sine of angle MCm = dt = Mn , x ( 1−tt ) thus the equation becomes

But since the body has fallen from the height DQ, then v = b + g .DQ = b + ga ( 1 − m 2 ) − gx ( 1 − tt ) . With which values substituted in the equation, this equation arises : bdx 2 ( 1 − tt ) = a 2bdt 2 + ga3dt 2 ( 1 − m 2 ) − ga 2 xdt 2 ( 1 − tt ) − bx 2dt 2 or Which equation thus by integration, as t = m makes x = a, expresses the nature of the curve sought. Q.E.I.

Corollary

  1. If in place of the sines of the angles ACD and MCD the cosine of these are introduced, these become … ( 1 − m 2 ) = n and ( 1 − tt ) = q , then we have .. or … which thus has to be integrated, so that on putting q = n , we have x = a.

Corollary 2

  1. Where the curve is normal to the radius CM , with dx vanishing there, it becomes : b( a 2 − x 2 ) = ga 2 ( qx − na ). Hence, whenever the curve is normal to the radius CM . Moreover since q is contained between the limits
  • 1 and – 1, x cannot be a greater than the quantity given ; for with x =∝ , q =∝ , which is absurd.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 194 Corollary 3.
  1. If CM is normal to the curve, then the angular speed is equal to which must be equal to b . Therefore that is the maximum angular speed if q = – 1. a Moreover that angular speed is made less than that, when x is made greater. Now again the angular speed is less than that, when the angle of the curve to the radius MC is greater. Whereby the curve cannot descend below a certain distance C, as the distance giving x is found from this equation : clearly, Therefore this is the maximum distance of the curve from the point C. Corollary 4.
  2. Therefore since the curve is not able to be at a greater distance from the fixed point C, [p. 127] this curve returns on itself. Clearly either after one revolution or after two or three etc., or even after an infinite number of revolutions it will return [to its starting conditions], as the letters a, b, n and g assumed. Example.
  3. If the force acting vanishes, then g = 0 and the body advances uniformly. Therefore this equation is then obtained to describe the curve : the integral by logarithms is : or Which on reduction gives : The line AC falls on the vertical CD; indeed likewise on account of the force g vanishing, this must become x = a with t = 0, from which there arises a 2 + c 2 = 0 andEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. 4 2 2 4 2 2 2 page 195 a = a x + a t or x = a ( 1-tt). Which is the equation of a circle of diameter a passing through the fixed point C. For when the motion is uniform on a circle, the motion is also uniform with respect to any point on the circumference.

Scholium

  1. Moreover it is evident in this case that the circumference of the circle is a satisfactory curve, the centre of which is at the fixed point C, which solution is Cleary the easiest and which can be produced spontaneously. On account of which it is a source of wonder that this case is not contained in the solution. [p. 129] Now the reason for this is clearly similar to that, as we deduced above (268), where we observed a similar paradox.

Having designated C as the centre of the circle, it is necessary to set x = a or dx = 0, which now, since x is considered as a variable quantity, is unable to be done, especially since in the same equation the solution is otherwise contained, in which x really is a variable quantity. Now from the first equation on putting v = b, which is Mn.a = Mm.x , it is understood that the circle can be a satisfactory solution ; for if x = a everywhere, then also Mn = Mm. Moreover I consider the great help to be given, in producing the construction of the curves satisfying this problem, if a solution could be found by such a method, which at the same time should give the case of uniform circular motion about a centre C. For as the most simple case to be present in the solution has thus been removed, as it cannot be found, we conclude that it is often the case with other curves that simple curves are contained in the solution of some general curves, which are hard to solve.

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