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Chapter 2

The Rectilinear Motion Of A Free Point Acted On By Absolute Forces

by Euler
March 20, 2025 31 minutes  • 6500 words

THE MOTION OF A POINT ON A GIVEN LINE IN A VACUUM.

PROPOSITION 31. Problem. 282. 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 )EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 193 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. Corollary1. 283. 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 [p. 127] which thus has to be integrated, so that on putting q = n , we have x = a. Corollary 2. 284. 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.
  4. 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.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 196 PROPOSITION 32. Problem.
  5. If the body is attracted by some force to the centre of force C (Fig. 37), to find the curve AM on which the body is carried with a speed equal to the uniform motion towards C. Solution. [p. 130] Let the minimum speed of the body at A correspond to the height b, the line CA is a tangent to the curve at A, since the body at A ought to move directly to C. Let AC = a and CM = x, the speed at M corresponds to the height v and the centripetal force at M is equal to P; then we have v = b − Pdx , as the integral has to be taken in order that ∫ with x = a it vanishes and makes v = b. Now the speed at M has to be of such a magnitude, since the element Mm is completed in the same increment of time as the element Pp with the speed b . Hence it becomes b : v = Pp : Mm = MT : MC , hence this equation is produced [for the triangles Mnm and MTC are similar] : The perpendicular CT to the tangent is equal to p; it becomes Or, if the sine of the angle ACM is put equal to t, it dt . Hence the following equation is produced mn = becomes [on differentiation]: MC CT .nM = [as mn = MT p .− dx ( 1−tt ) bp 2 ; and x 2 − p 2 = Pdx , etc. ] : 2 2 ∫ x −p Of which for each case, if P is given in terms of x, a suitable curve can be constructed. Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 197 Corollary 1.
  6. If the centripetal force is proportional to some power of the distances, clearly n P = xn , f then we have : ∫ Pdx = (xn+1−)af , n +1 n +1 n With this substituted we have the following equation for the curve AM : [p. 131] Which equation must thus be integrated, so that with t = 0 it becomes x = a. Corollary 2.
  7. If ∫ ( 1dt−tt ) is thus taken, so that it becomes equal to zero if t = 0, there is produced by integration from that equation: if with the centre C and with the radius BC = 1, the arc of the circle BSs is described. From which it is apparent that the curve AM has an infinite number of rotations before it arrives at C. For with x = 0, BS is made ∝ . Corollary 3.
  8. Therefore the construction of this curve depends in part on the quadrature of the circle and in part on logarithms, if n + 1 is a positive number. But if n + 1 is a negative number, that term which was given by a logarithm, is also reduced to the quadrature of the circle. Corollary 4.
  9. This curve has a turning point [flexus contrarii]where dp = 0. [p. 132] Therefore in order that this point can be found, the equation is taken ; from which by differentiation, and on putting dp = 0, there is produced :EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 198 Corollary 5. n
  10. Therefore in the case in which P = x n , the turning point is at the place where : f Thus, this equation arises: Which on substitution into the integration gives the angle ACM at which the turning point lies. Scholium 1.
  11. But since from the nature of these kinds of curves it is difficult to produce turning points in general, it is considered that the principles are best effected by descending to the level of special cases. Example 1.
  12. If the centripetal force is proportional to the distances or P = xf , in making n = 1. Therefore with the arc BS put equal to s, the curve sought is expressed by this equation : [p. 133] From which equation with any distance of the point M from C the angle BCS is found, at which the body is without doubt present at that distance. Now between the distance MC = x and the perpendicular CT = p, there is this equation : The turning point of this curve is where dp = 0, but this is where or since x cannot be greater than a. Hence it becomes : andEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 199 Therefore the cosine of the angle that the curve makes at the turning point with the radius CM is equal to [Euler’s original formula has here been corrected by P. S. in the O. O. to give the one presented here] : Now the equation of the curve converted into a series on putting ( a 2 − x 2 ) = y is this : Therefore in this series, the beginning of the curve, where x is not much less than a, or y is extremely small, will be Then from this equation it is apparent on making x = 0 that s becomes ∝ , whereby the curve goes round the centre C in an infinite number of spirals, and when the body is near to the centre , the equation becomes : From which it follows that near the centre C the curve goes off in a logarithmic spiral. Example 2.
  13. Let n = – 1 or n + 1 = 0, which case it to be extracted from the differential equation. f For it becomes, on account of P = x thus this equation is obtained : [p. 134] the integral of which is : The other equation between the perpendicular p and x is given by :EULER’S MECHANICA VOL. 2. Chapter 2d. page 200 Translated and annotated by Ian Bruce. From which the turning point is found at that place, where or Therefore by taking this expression, there is obtained : Again it is evident, if x is made equal to 0, that s =∝ or the curve makes an infinite pp number of turns around the centre C ; now in this case we have xx = 1 or p = x. Therefore finally the curve goes round in an infinitely small circle. Example 3.
  14. Putting n = – 2, in order that the centripetal force is inversely proportional to the square of the distances, the equation becomes Now ∫ 1+ yy expresses the arc, the tangent of which is y or a −x x ; let this arc be equal to dy t, and the equation becomes : is Therefore everywhere with the distance given x the arc s taken and multiplied by 2ab f equal to the difference between the tangent a − x and the corresponding arc with the x radius put equal to 1. If x is put equal to 0, then this makes s =∝ , from which it follows that the curve falls towards the centre C in an infinite spiral. [p. 135] Besides, on account of it follows thatEULER’S MECHANICA VOL. 2. Chapter 2d. page 201 Translated and annotated by Ian Bruce. pp From which it follows if x vanishes that xx = 1 or the curve finally also goes in an infinitely small circle. If ab = ff, then The turning point in this case lies at that place where 2ax = 3 xx , where either x = 0 or x = 23a . But if it is not the case that ab = 4ff, then and the turning point is found by taking [either x = 0 or] x = a 13 . Scholium 2.
  15. Moreover from which it is apparent how the infinite spirals can be compared, if the n centripetal force is proportional to some power of the distance, or P = x n , and the f equation between p and x is considered, which is Here two cases are to be distinguished, the one in which n + 1 is a positive number, and the other in which it is negative. If n + 1 is a positive number, on making x = 0 the equation becomes Hence in this case the curve AM goes around the centre C in a logarithmic spiral. pp But if n + 1 is a negative number, on making x = 0 the equation becomes xx = 1 . Therefore in these cases the curve at C becomes an infinitely small circle. The speed of approaching C in these cases becomes infinitely large, and on account of this, unless the curve becomes a circle [p. 136] the body approaches C with an infinite speed, which is contrary to the condition of the problem. Therefore with the curves determined, upon which the body uniformly approaches the centre of forces, we will investigate these curves in which the body is carries around the centre of force with a uniform motion.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 202 PROPOSITION 33. Problem.
  16. If a body is always attracted to the centre of force C (Fig. 38), to determine the curve AM, upon which the body is moving with a uniform angular motion around the centre C. Solution. Let A be the highest point of the curve, where the curve is normal to the radius AC, and let the speed of the body at A correspond to b and AC = a; the angular speed at A = ab , to which quantity the angular speed at some particular point M must be equal. Putting CM = x, to which CP is taken equal, and the centripetal force at M is equal to P; then the speed at M corresponds to the height b − Pdx , ∫ ∫ with the integration of Pdx thus taken, so that it vanishes on putting x = a. With the tangent MT drawn, the perpendicular sent from C to that is called CT = p; then we have x : p = Mm : mn. [p. 137; again, the infinitesimal triangle Mnm and the finite triangle MTC are similar] On account of this, the speed passing along mn is equal to and the angular speed is equal to which must be equal to ab . Hence the following equation is produced : orEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 203 With centre C and with radius BC = 1 the arc of the circle BS is described, which is called equal to s, then 1 : ds = x : mn , and hence mn = xds and Mm = ( dx 2 + x 2ds 2 ) . Since now we have x : p = ( dx 2 + x 2ds 2 ) : xds , the equation becomes With which value substituted in the equation found gives and hence From which equation the curve sought can be constructed. Q.E.I. Corollary 1. ∫
  17. When x is made smaller, then the larger b − Pdx becomes, whereby when x is p made smaller, with that also x becomes less, or the sine of the angle CMT. For p x b = . x a ( b − ∫ Pdx ) Corollary 2.
  18. Again, as by hypothesis since in this equation x cannot become greater than a ; for it would make p > x. On account of which none of the radii CM can be normal to the curve, unless it is at a maximum, clearly equal to AC. Scholium 1.
  19. Indeed it is evident through these that for whatever hypothesis of centripetal force, it is satisfactory for a circle with centre C to be described [p. 138] ; for the body must be moving uniformly on a given circle. Moreover even if the general equation does not seem to include the circle, yet no less it must be contained, as we have now intimated above. [Part of Euler’s problem was the imperfect state of affairs at the time regarding dynamics : we would now make some reference to angular momentum, which he was later to clarify, in the analysis. This of course makes these special curves all the more fascinating for us to look back on; thus, Euler’s use of a potential energy or a work related function was introduced as a means of simplifying problems – only to be discovered later that this was how the world really worked, although Euler understood that he had to base his dynamics on known physical facts, such as Huygens’ pendulum as a means of finding the acceleration of gravity, and Galileo’s inclined plane : it was the latter that gave rise toEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 204 the potential energy function that relates height fallen from rest to the square of the final speed.] Scholium 2.
  20. Moreover it is evident that no other curve going around the centre except the circle is able to satisfy the condition sought. For in curves of this kind it is not possible that all the lines drawn from the centre and normal to the curve are equal to each other. Therefore which curves besides the circle which solve the problem, these must pass through the centre C itself, so that not more than one radius MC is normal to the curve. These curves are of this kind that we look at in the following example. Example.
  21. Let the centripetal force be directly proportional to the distances from the centre or P = xf ; the equation becomes : With which substituted, the following equation is produced for the curve : [p. 139] Now the arc is ∫ ( adx− x ) , the sine of which is ax with the total sine arising 2 2 equal to 1. This arc is denoted by A. ax . Let the sine of the arc BS = t, then s = A.t; hence the equation becomes : Or the arc, of which the cosine is ax , is equal to : Hence the construction of the curve easily follows on, and it is an algebraic curve, whenever a 2 + 2bf is a rational number. Let 2bf giving the integral of this, by means of imaginary logarithms, is given by :EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 205 or The perpendicular MQ = y is sent from M to AC and on putting CQ = u there is y 1 : t = x : y and so t = x . Hence, 2 Or, if m = 2 or bf = a6 , then this equation is obtained : which reduced gives : or But if an equation is desired between the orthogonal coordinates u and y, then it is this equation of the sixth order : In this curve the applied line is at a maximum if x = 2b or if we take 3 for then it becomes: Now in other examples of this, with the values m , the maximum applied line is whereEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 206 PROPOSITION34. Problem.
  22. Let the uniform force acting be g pulling downwards everywhere and the given curve AT (Fig. 39) [p. 140] ; to find the curve AM, upon which a body thus descends, so that the time to pass through any arc AM is proportional to the square root of the corresponding applied line PT of the given curve AT. Solution. The common abscissa AP = x, the applied line of the curve AT is PT = t; hence the equation between x and t is given, since the curve AT is given, which must be such that with x = 0 makes t = 0 also, since the initial motion is put at A and the times reckoned from the point A. Again let the curve sought be AM with the applied line PM = y and the arc AM = s. Let the initial speed at A correspond to the height b. Hence the speed at M corresponds to the height b + gx and the time in which the arc is completed is equal to which must be equal to t . Hence this equation is obtained : Hence and From which equation, since t is given through x, the curve sought AM can be constructed. Moreover this has to be constructed so that x = 0 also gives y = 0, as the start of the curve AM is at A. Q.E.I.EULER’S MECHANICA VOL. 2. Chapter 2d. page 207 Translated and annotated by Ian Bruce. Corollary 1.
  23. Therefore in order that the curve is real, it is necessary that bdt 2 + gxdt 2 shall be greater than 4tdx 2 , or [p. 141] For if the relation becomes then the curve AM becomes a vertical straight line, upon which the quickest descent is made. Corollary 2.
  24. Therefore if the curve AT somewhere makes dt equal to 2 t dx , then the tangent b + gx there corresponding to the curve AM is vertical. And if beyond this point it satisfies dt < dx then the curve AM does not descent to that point, but has a turning point at 2 t b + gx that point where the tangent is vertical. Corollary 3.
  25. If the angle that the curve AT makes to the vertical AP at A is acute, the tangent of which is m, then at the beginning A : hence m( b + gx ) must be greater than 4x, which always happens if b is not equal to 0. Moreover it then becomes : dy Therefore with x = 0 , dx =∝ , or in these cases the tangent to the curve AM at A is horizontal, unless b = 0. But if b = 0, then it becomes Therefore least the curve AM becomes imaginary, gm must be greater than 4 and then the curve AM makes an acute angle with AP at A, the tangent of which is ( gm − 4 ) . 2EULER’S MECHANICA VOL. 2. Chapter 2d. page 208 Translated and annotated by Ian Bruce. Corollary 4. [p. 142]
  26. Now if the angle that the curve AT makes with the vertical AP at A is right, then m =∝ . Therefore in this case the tangent to the curve AM at A is always horizontal, then either b is made equal to zero, or otherwise. Corollary 5.
  27. If the speed at A is equal to 0 and at the start A the curve AT is combined with the curve, the equation of which is t = αx n with the number n taken as positive, so that as x increases, so too does t, then we have : Now least dy becomes imaginary on making x = 0, it must be true that n > 2n − 1 or n < 1, in which case clearly the curve AT is normal to AP at A. For now at the point A, n 2αgx n and the radius of osculation of the curve AM at A is equal to 2( n −1 ) . From which it follows for the curve AM, the tangent of which is horizontal at A, that the radius of osculation at A must be infinitely small, if the body is able to start from rest on that curve. For unless the radius of osculation is infinitely small, the body remains at rest at A for ever. Corollary 6. [p. 143]
  28. Therefore if the body placed at A descends from rest, so that the curve AM is made real, then dt must be greater than dx , even at the start of the curve AT. Whereby if we 2 t gx put where p is a positive quantity, even if x is made exceedingly large, then we have : where ∫ pdx must thus be taken, so that it vanishes on putting x = 0. Moreover with this value in place, on substituting for dt , there is produced 2 tEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 209 for the curve AM sought. Or this equation is obtained between x and y : dt = 1 + p into the above expression for dy [This follows directly by substituting 1 dx gx 2 t 2 dy dt ⎞ − 1 and integrating.] written in the form : dx = gx⎛⎜ 1 dx ⎟ ⎝2 t ⎠ Now it is to be noted that p cannot be such a quantity that ∫ pdx can be made infinitely large when integrated in the prescribed manner. Corollary 7.
  29. From what has been said it is understood that as long as the value p is kept positive, the body descends the curve AM ; if we male p = 0 and then negative, then the curve has a cusp at that place and returns up again. If p =∝ with pdx yet remaining finite, then ∫ the curve AM has a horizontal tangent there. Corollary 8.
  30. If b is not put equal to 0, from the same curve AT innumerable curves AM can be found; since the initial speed can indeed be taken as greater or less, with another curve AM produced. Scholium. [p. 144]
  31. The greatest use of this problem is in the solutions of the following indeterminate problems, in which all the curves are required, upon which a body in the same time arrives at either a given straight or curved line. On account of this we will investigate the innate character of the quantities t and p with more care, so that we are allowed to use these in the following propositions.EULER’S MECHANICA VOL. 2. Chapter 2d. page 210 Translated and annotated by Ian Bruce. PROPOSITION 35. Problem.
  32. With a uniform force acting in the downwards direction, to find all the curves AMC (Fig. 40); upon which a body beginning to descend from rest at A, arrives at the horizontal line BC in a given time. Solution. Putting AP = x, PM = y and AB = a. In the curve AND, PN expresses the above assumed quantity pdx , ∫ and a property of this curve must be that it meets the axis AB at A, and that the applied lines increase on being continued even as far as D, so that clearly pdx is positive. Now by taking (312) the time to traverse AMC = 2 ga
  • BD . [From g dt = dx + pdx on integrating. Do realise that the 2 dt gx curve in Fig. 40 is no longer the given curve AT in Fig. 39, which is still part of the calculation, but the extra arbitrary component for which NP = pdx .] On account of ∫ which, since an infinity of curves of this kind can be substituted in place of the curve AND, from these an infinite number of curves AMC can arise, upon all of which a body reaches the horizontal line BC in the same time from A [p. 145]. Therefore in order that this may be obtained, such a quantity must be taken for pdx , which vanishes when x = ∫ 0 and which becomes equal to BD on putting x = a, with p retaining a positive value everywhere along AND. Q.E.I. Corollary 1.
  1. If on making x = a, p = 0, or if the curve AND at D stands perpendicular to the horizontal CD, then the curve AMC also remains perpendicular to DC. Corollary 2.
  2. And if on putting x = 0 also p = 0, the tangent to the curve AMC is vertical at A ; now likewise also it comes about if p x becomes equal to zero on putting x = 0. But if p x becomes infinite on putting x = 0, then the curve AMC has a horizontal tangent at A.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 211 Scholium 1.
  3. Therefore the problem is understood to be indeterminate to a large extent, since in an infinite number of ways an infinite number of curves AMC can be found. On account of which in the following examples we indicate however many series of the infinite curves sought satisfying the question that we have been pleased to find. Example 1. [p. 146] ∫
  4. Putting PN = pdx = z and BD = b , thus in order that the descent time must be equal to 2 ga

g b . For the curve AND, this equation is taken: z = αx 2 + βx , which ∫ now has this property, as pdx or z vanishes on putting x = 0. Now since on making x = a we must have z = b , there is obtained b = αa 2 + β a and hence β = ab − αa and thus z = αx 2 + x a b − αax. dz must always have a positive value, if x < a, it is necessary that Then since p or dx 2αx + ab − αa is positive. Whereby it is required that putting b = αa 2 + αaf , then α = z= b > αa 2 ; therefore on b . With which substituted, there is obtained a 2 + af x 2 b + fx b , which equation, with innumerable positive values substituted in place of a 2 + af f , gives the curves AND. Moreover there becomes : from which it is apparent that all the curves AMC hence arising are tangents to the line AB at A. Now the equation for the curves AMC is this [from (312)] : Which contains an infinitely of curves satisfying the problem, upon which the descent time for all to the horizontal line is equal to 2 ga + g b.EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 212 Corollary 3. 321. Now all these line are rectifiable. For since [p. 147] then we have Now Thus the total area under the curve AMC is equal to : Corollary 4. 322. Therefore among these curves AMC the longest is produced if f = 0; for then AMC = a + 54 gab. And for this, that equation becomes : Now the shortest is obtained by making f =∝ ; for then it becomes AMC = a + 23 gab. And the equation for this curve is : Scholium 2. 323. All the curves AND contained under the equation are parabolas, thus so that by parabolas alone innumerable curves are found satisfying the problem. Now neither are all the parabolas contained by this equation, as in place of this equation, if this other equation is used :EULER’S MECHANICA VOL. 2. Chapter 2d. page 213 Translated and annotated by Ian Bruce. which also contains an infinite number of parabolas, again an infinite number of curves AMC are found, upon which a body completes the descent in the given time. From which it is to be understood that if only conic sections are substituted in place of the curve AND, then such an infinite number of curves AMC can be found. [p. 148] For taking this equation for the curve AND : z 2 + αz = β x 2 + γx + δxz , which contains all the conic sections passing through the point A, then it must become [for BD]: b + α b = βa 2 + γa + δa b γ and both α and β a + γ +δ b must be positive quantities; which is easily seen, since it can α + 2 b −δa be done in an infinite number of ways. If then all the algebraic curves are considered and afterwards also the transcendental curves likewise, the greatest wealth of all the curves described in the same way con be conceived. Example 2. n 324. This general equation z = x n b is taken for the curve AND with n denoting some a positive number; z vanishes on putting x = 0 and the equation becomes z = b on putting n −1 b dz = nx x = a, as it is requires; now besides also the quantity p or dx is positive. n a Therefore since [the appropriate quantity] becomes p gx = nx n− 1 2 a gb n , then the equation for y is : Which equation includes an infinite number of curves AMC, which are all rectifiable. For the arc becomes : and thus Corollary 5. [p. 149] 325. If n = 12 , then the equation becomes andEULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. page 214 Whereby the curve becomes an inclined straight line, upon which the descent is in a time 2 ga equal to g + b . Therefore it is evident the shorter lines [i.e. curves] are given by this inclined straight line, upon which a body from A arrives at the horizontal BC in a given time; indeed on making n < 12 the line AMC becomes shorter. Scholium 3. 326. Otherwise if a single curve AND is given, and it is required to provide a curve AMC, from that curve itself innumerable others can be found. For with one equation given between z and x taking hence with diverse values of m, innumerable curves are found. In a similar manner also we can put : for it becomes PN = z = b , on putting x = a. And in general, if P is any function of x and z, A is now the same function that is produced on making x = a and z = b , and it can be taken that PN = Pz . Moreover P must be such a function that Pz vanishes on A making x = 0 and z = 0, and the differential of PN divided by dx must be a positive quantity, for as long as x < a. Scholium 4. [p. 150] 327. In a similar manner the most general problem can be solved, if on designating P as some function of x vanishing if x = 0, and A is that quantity which P becomes if x = a, on taking z = P Ab , for the most general equation for the curve AND. Then we have Q b dP = Qdx, Q must be a positive quantity, as long as x is not greater than a ; then p = A and hence which is the most general equation for the curves AMC, which are all completed by a body descending in a proposed time. It is apparent in this way that transcendental curves can also be substituted in place of the curves AND, in which cases the time, in which some part of the curve AMC is completed, cannot be defined algebraically. If Q gbx is put equal to R, then we have :EULER’S MECHANICA VOL. 2. Chapter 2d. Translated and annotated by Ian Bruce. Therefore with some function of x taken in place of R in order to find A, page 215 Rdx must be gbx integrated, so that it thus vanishes on putting x = 0; then it is required to put x = a, and as that comes about, it is equal to A. Now here only this has advise has to be given, that a positive quantity is taken for R, for as long as x does not exceed a, and it must be warned that Rdx should not becomes infinite, if the integral is taken in the prescribed manner. ∫ gbx

The Rectilinear Motion Of A Free Point Acted On By Absolute Forces
The Rectilinear Motion Of A Free Point Acted On By Absolute Forces

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